1. 1-1/2" thick gypcrete should be ~14 to 16 psf, if I recall correctly. Prolly closer to 14. Regardless, that is very dense when it comes to residential floors that are not on grade. That's good for this purpose, though concrete tends to conduct LF fairly well (a minus). With an air gap above a double-DW ceiling in your HT room, plus some insulation in there.... That ought to work well. Plus how often do you use your dining room anyway? Most people hardly ever use it.
2. My comments about speakers and their audio dispersion (polar plots) don't have anything to do with the ports. What I was referring to is every speaker has a pattern with which it disperses sound - in a 360 degree arc on each axis (x, y, z) - creating a sphere of sound projecting outward. The most intense/loudest sound comes of course in the front, where the speaker's sound-emitting portions are pointed. However, all speakers "leak" sound in other directions as well, such as from reverberations of the speaker cabinet, and just plain the way sound travels. My point was simply that some speakers are better than others at directing as much as possible of the sound/noise forward. Some are very "leaky" per se and tend to pump out more sound at different frequencies at other angles. You want to try and avoid the latter. The only way you'll know, however is by looking at polar plots.
And I want to add that this is really a micro attention to detail for most people. If you had your speakers right up against the wall, it might be a bigger issue. The only reason I mentioned this is because as I said above, the angle of your wall behind the front side speakers is going to direct those reflected sounds at unusual angles into the room. It could theoretically be something you want to deal with. However, if you have a lot of absorption behind and to the side of those speakers, I believe you'll have no issues. You may wish to consider using a baffle wall, as an example.
3. Is 5/8" better than 1/2" drywall? Yes. Is it a HUGE difference? No. You're talking less than a handful of STC points. That said, the more layers, the more the difference is amplified. If you don't need absolutely the best sound proofing then you'll likely be fine with the 1/2". I understand the motivation of not moving 5/8" sheetrock.
The key is really to use a dampener in between - a viscoelastic compound; e.g. Green Glue, Quiet Glue, TiteBond, etc. (btw, don't use TiteBond).
Presuming a 2x4 wall cavity with wood studs, insulation in the walls, and the sheetrock attached directly to the walls (i.e. no clips & channel, etc.), the delta between 1x 5/8" sheetrock and 2x 5/8" sheetrock with no glue in between is 2 or 4 STC, depending on your source of info (i.e. who's test you're reading). The delta between 2x 5/8" with no viscoelastic glue and 2x 5/8" DW with Green Glue in between is 7-11 STC. If you use clips & channel + viscoelastic glue, the delta jumps to 20-23 STC (a SUBSTANTIAL difference).
4. I would recommend clips & channel at least on your common (interior) walls. Ideally, you'd have either a) complete room-within-a-room (double stud wall and independent [floating] ceiling); or b) clips & channels on all walls and ceiling. That would be your best isolation. If you like to play movies and music loud and your significant other hates that, this would be your best bet (including viscoelastic glue in between 2x DW layers on all surfaces). You could still skimp a bit and go with 1/2" sheetrock. The isolation offered by the clips & channel + viscoelastic glue will do more for you than the extra mass of the 5/8" drywall (though mass is a big helper in absorption).
If you don't blare your content super loud all the time, you could compromise and use more isolation for those common walls. It would save you time and money. Just depends on your priority. The biggest factor is it's obviously very difficult to re-do anything in the future if you feel you didn't do enough the first time around.
Many people on this forum will tell you to goto the max with isolation, but IMHO one needs to be practical if you're not an ADHD audiophile and this is your home. Like I said, it all depends on your priorities and needs/wants.