Glasperlenspieler

1. I'd say it's generally a bad idea to publish a book before you have your Ph.D.
2. Getting a good grade on something doesn't necessarily mean the professor thinks it should be published.
3. For many (most?) fields it's not necessary to have any publications to get admitted to a good Ph.D. program.
4. At this stage of your career, all publication decisions should be made in close coordination with a knowledgeable and sympathetic advisor (random people on internet message boards don't count).

Even schools with rolling admissions have a priority or initial deadline, and as long as you have your application in by then, it shouldn't make a difference.
I think it's usually good to submit a couple days before the deadline in case unexpected problems arise, but nobody will be looking at your application until after the deadline.

To preface my advice, I'd like to mention that programs in general are admitting smaller cohorts this year. I wouldn't be surprised if this trend continued beyond 2021. If you're thinking about PhD programs with the job market in mind, you'd be better served by focusing your efforts elsewhere. It was already near impossible to land a tenure track position. Now, things are looking very, very bleak. 
The best thing you can do is refine your interests and develop questions about them. You aren't expected to be an expert in a field when you apply, but you should be somewhat conversant with one. Write a good (that is, relevant to your field) thesis and use it for your writing sample. 
Conferences and publications don't hurt your application, but they take time. The same is true for library work and volunteering. In my opinion, you're better off spending your time researching and writing. Develop relationships with faculty to help you along. 
The application process is expensive and grueling. I'd heavily consider whether you want to pursue this route, because a lot of the labor that goes into making yourself an ideal PhD applicant may also prevent you from developing your marketability in other avenues. 

Even schools with rolling admissions have a priority or initial deadline, and as long as you have your application in by then, it shouldn't make a difference.
I think it's usually good to submit a couple days before the deadline in case unexpected problems arise, but nobody will be looking at your application until after the deadline.

This coming app cycle is already fucked, FYI.

These are good questions for your advisor/a professor in your field whom you trust.

It's unusual, but not unheard of, depending on the professor and relationship to the student [I've had similar experiences]. Given what @wwfrdhas said about this professor's stature in the field, I'd still use him as a recommender. At absolute worst, it sounds a little bit like he's guarding his own turf and wants to 'tag' the OP as one of his students, which is complimentary if not territorial. And that's probably not even what's going on; it's simply my most cynical interpretation. Given his age, it's more likely a generational thing. 
I'll admit that I preferred having my professors upload their letters directly, but that isn't always feasible, especially when applying to a great many schools. For some universities/committees, it seems not to matter much at all whether a letter comes from Interfolio or the professor themselves; others will outright refuse to accept Interfolio letters.
If this professor has given you a 5-university cap [in terms of individually-uploaded letters], I suggest making sure you direct him to those universities that won't accept Interfolio letters; then, if there are any left over after that, have him do a direct upload to your first choices among those. 

Just to respond briefly to this part: it’s not clear to me that this is all that unusual. Most professors that I had, as an undergrad, from whom I took upper-division courses in my major would say something at the end of the course along the lines of ‘should you ever need a recommendation, I’d be happy to’. It is my understanding that some professors may make such an offer to any Of their students for whom they feel they could write a good recommendation. 
 
However, I agree with the rest of your advice, namely, that OP ask their other letter writers about the situation. 
 
editing to add another thought: There were a few places where I knew one specific professor’s recommendation would help a LOT. So I had one letter-writer send their letters to just a few places. Perhaps this professor has something similar in mind. My advice for OP a would be to show the professor the list of programs and ask, directly, ‘where is your letter likely to help me the most?’. Then have them send the letter to those programs. 

I think a lot here depends on one's orientation towards those topics as well as what you mean by continental vs. analytically oriented departments. When I first heard that constellation of interests, I immediately thought of Chicago and Pitt (though I can't imagine McDowell and Brandom will be around for all that much longer). I would also think about Berkeley, Brown, JHU, Riverside, and maybe Stanford. Or at the very least those are the schools I would be thinking of given the way I tend to approach those fields. Now maybe you were thinking of a place like Chicago as a continental school. Maybe, but I tend to think there's a pretty big divide in approach between these sorts of schools and the SPEP crowd.

The MA route sounds like a much better option, but better to do it in the US or Canada where you can get funded

Yeah, that's what I mean. I would think those are programs where those kinds of interests could be supported. It's not because they're analytic, though; it's because they have a strong historical contingent which includes those periods/movements. Otherwise, as far as I know, those interests (apart from Kant) are mostly only alive at straight-up SPEP programs.

I think a lot here depends on one's orientation towards those topics as well as what you mean by continental vs. analytically oriented departments. When I first heard that constellation of interests, I immediately thought of Chicago and Pitt (though I can't imagine McDowell and Brandom will be around for all that much longer). I would also think about Berkeley, Brown, JHU, Riverside, and maybe Stanford. Or at the very least those are the schools I would be thinking of given the way I tend to approach those fields. Now maybe you were thinking of a place like Chicago as a continental school. Maybe, but I tend to think there's a pretty big divide in approach between these sorts of schools and the SPEP crowd.

For me, 0$. I would NOT have pursued the MA/PhD if I was not fully funded. It’s not worth it - the time it will take to get you out from under that debt considering the academic job market just won’t add up. Especially now since we have no idea what the job market will be looking like the next few years. 

You should make sure that a) your writing sample engages with the secondary literature written by people at the program's you're applying to and their interlocutors and that you situate what you're doing in relationship to that scholarship, and b) you should make sure that your paper emulates that scholarship in terms of style, format, approach, etc.

You should make sure that a) your writing sample engages with the secondary literature written by people at the program's you're applying to and their interlocutors and that you situate what you're doing in relationship to that scholarship, and b) you should make sure that your paper emulates that scholarship in terms of style, format, approach, etc.

Don't play the game of trying to predict which areas will be hot when. It's not a game you can win, except by luck. And it's not a game you'll be well-equipped to play until the end of your PhD at the earliest, when you're much more familiar with the discipline, your fields, have been on the market a number of times, have done the conference circuit a bunch, etc. (Besides which, most hot areas are only briefly hot--that is, they heat up for a few years then hiring peter out because the demand has been mostly filled). Cultivate a diversity of interests, and follow those interests. It's standard for people now to specialize in two (sometimes three) distinct areas; so cultivate your two. Where planning is concerned, try not to cultivate two areas which are dead ends jobs-wise (e.g. aesthetics and 19th-century Germany). But beyond that, just go for what interests you. You won't get a job anyway, so you might as well enjoy it as much as you can!

Many departments provide profiles for their grad students which include their educational background. Yale's Spanish department, for instance, does this: https://span-port.yale.edu/people/graduate-students
Spend some time looking at departmental webpages and you can get a decide idea of the profile for a typical student.

@tmck3053, I think we're pretty much on the same page. It's definitely an absurd game, and I don't want to defend it. Just trying to lay out the contours to the degree I understand them. I think you're right though that there are some Ivy-plus institutions that will hire people with perfect pedigrees and no publications but look down upon someone with a publication in a lower tier journal. I think publishing strategy is going vary depending on what sort of institutions you're getting your PhD from and what sorts of institutions you're applying to (and maybe subfield as well). I stand by my " if as a BA or MA student you can get published in a top tier journal, great. But if not, don't worry about it, sit on it, and just try to make it the best writing sample you can." But yeah, a lower tier publication definitely isn't going to sink your application, but I don't think it'll help all that much and there may be reasons to wait and see if you can do something more with it down the line. (The exception to this is that some subfields are really fast moving, such that a response paper may be very timely now, but maybe no one will care in 4 years. Idk, it's complicated).