First, congratulations on the coveted HLS acceptance, whether it's what you want to pursue or not.
1. As a person who's researched joint MPP/JD programs, it's pretty standard for admissions decisions for both schools to be independent and unrelated. Getting accepted to HLS will not, on its face, give you an advantage when applying to HKS. I will say, however, that the qualifications that got you into HLS are likely to be pretty appealing to adcomms at HKS as well.
2. Most schools give averages or ranges for the GRE scores of previously admitted classes. On the whole, I'd say if you score in the 75th-80th percentile or above, you'd be in a pretty good spot, especially considering your high GPA and verbal/writing skill as evidenced on your LSAT performance. Get a 160+ on the Quant and you'll be well-situated.
3. HKS is not rolling, nor are most MPP/MPA programs' admissions processes. The deadline for HKS is the beginning of December (December 2nd this past cycle), so get your application in by then and your application will be considered along with everyone else.
4. From the HKS website (emphasis mine): "HKS is a school of leadership and public service. Successful candidates typically demonstrate leadership experience and commitment to the public good, which are qualities often displayed through professional experience and volunteer work during and after college. Our curriculum is demanding, and we look for applicants with the skills needed to manage rigorous coursework. Course requirements vary by program; however, quantitative coursework is part of the curriculum for all programs. Previous coursework in micro- and macroeconomics, and multivariable calculus is required for the MPA/ID Program. Coursework related to quantitative methods (e.g. economics, mathematics, and statistics) is recommended for our other degree programs. Applicants lacking quantitative coursework are strongly encouraged to take these classes prior to applying. Successful candidates also demonstrate strong writing and communication skills. Work experience is preferred, and some degree programs have specific work experience requirements. Read more about our degree programs for more information."
Looks like these courses are not required, but rather recommended (and will probably help you be more competitive). I'd recommend you take Statistics/Microeconomics in the summer or the fall, especially if you're worried about your Quant GRE. When applicants lack the coursework, I've read that adcomms look to the GRE Quant score to see indicia of quantitative ability.
Hope that helps, and lastly, coming from someone whose dream would be to study Constitutional law from HLS' legendary faculty, don't pass on this tremendous opportunity until you fully consider what is right for you and what you want to achieve. That said, I wish you all the best in pursuing a masters program in Public Policy. I myself am waiting on admissions decisions due to come in a few weeks. Cheers!