It's great that you are asking these questions early on. There are definitely certain things I would have loved to know at your stage.
In clinical psychology PhD and PsyD programs, one of the final requirements for degree completion is a pre-doctoral internship. During the course of the program, you will have clinical experiences that will help you narrow down the specific populations/settings you would like to eventually work in, as well as let you develop your clinical skills. This all culminates into a final, year-long paid internship (which you want to be APA-accredited as well). These internships are unaffiliated with your program, and you must apply through a process run by APPIC. This process is similar to the med school residency match. You apply towards the end of the program and then interview at places. An algorithm than matches you to a site based on your rankings and the site's rankings. The hours of clinical experience you gain in this internship are your pre-doctoral hours.
If you go to an APA-accredited or PCSAS-accredited clinical psychology program, you will be fine. There is some wiggle room in going to a non-APA and non-PCSAS accredited program, but you have to jump through more hoops and will not be able to work in many major hospital systems, such as the VA, or apply to the APPIC match and therefore not get an APA-accredited internship. You can find a list of all the APA-accredited sites here: https://www.accreditation.apa.org/accredited-programs
PCSAS is a newer accreditation, and as of now, most programs with that accreditation still have APA-accreditation, but some, like UC Berkeley, are intending to let their APA-accreditation lapse when the current status expires.
At the doctoral level, an internship refers to that formal process I discussed in my response to question 1. At the undergraduate level, that is still a training experience you get outside of your institution, but it is far less formal. A fellowship is a form of funding. Research is...research. It is the process by which scientists (psychologists included) try to answer questions related to their topic of interest. Research typically falls into 2 categories: basic and applied. Basis research is finding answers for the sake of finding answers, while applied research aims to answer questions within the context of a real-world application. Ideally, you would want to get involved in a research lab ASAP when in undergrad to gain more experience in the process.
There's an app for that. Some people may use Excel spreadsheets and such, but many use a website called Time2Track. You basically just input what you did on certain days. This will all be explained in the doctoral program.
Yes, you typically get help from multiple people. First off, there is a directory that lists all of the internships available and explains all the specific training opportunities. All clinical programs have a Director of Clinical Training who oversees the clinical training of students in the program. Your advisor/mentor will also assist you. If your program is a 4+1 (4 years on campus and 1 year of internship), you will be applying in the beginning of 4th year, so you have a long way to go before thinking about internship.
Not all, but most. Also, not all funded programs mean fully funded, where you get tuition remission and a stipend in exchange for being a TA, RA, or working in some sort of administrative capacity (such as admin in the psych department). You will have to look at the specific programs to see what they offer. It's a safe bet that you will be fully funded at an R1 institution. and a pretty fair bet at an R2. You really have to look at the program to see the specifics.
This is going to be based significantly on research fit. While in undergrad, try to get a sense of what you are really interested in studying further within the realm of clinical psychology. If you can gain research experience in that particular niche while in undergrad, that's awesome. If not, that's ok. You just need to be able to explain how what you have done connects to what you want to do. Also, it's ok if you change your mind halfway through undergrad. Additionally, while you want to practice in California, I highly suggest you do not limit yourself geographically to California-based schools. A. programs can tell if you are just applying because you want to be in the area vs. you have an actual interest in the school. B. Programs in desirable areas (such as California or NYC) have the most applicants, so they are even more competitive due to the geographic desirability. Also, your graduate stipend will go a lot further for living expenses elsewhere, so your quality of life will probably actually be better. You can always come back to California after you finish.
Get involved in research and get good grades. When doing research, try to get products (posters, maybe even publications). Graduate schools like to see that you've presented research at a conference (local/regional are ok, national are better). This shows them that you were able to see a project through from start to finish and were more involved than just doing data entry. However, when you join a lab, you will be doing a lot of the grunt work from the start. Be reliable and show initiative, then you may get more complicated tasks. Also, working with a faculty member in research will help you get much stronger letters of recommendation. If you can do an undergraduate thesis, I would also do that. The last part is doing well on the GREs.
Research. Research. Research. Maybe work at a crisis line for a bit. The types of clinical experiences available to undergraduates aren't really that clinical. The closest is the crisis line (suicide, domestic violence, at-risk youth, etc.).
Hopefully I've answered all of your questions. Also, this is a great resource to read: http://mitch.web.unc.edu/files/2017/02/MitchGradSchoolAdvice.pdf
Also, take this time to explore your interests. It's great that you have this goal now, and work towards it, but don't put the blinders on to other potential career paths as well. Think about why you want to be a clinical psychologist and think about other potential careers that also may offer similar benefits to what you want your day to day to look like.