First, are you sure that 30A is a sufficiently large breaker for your range circuit? Most electric cooking appliances (freestanding electric ranges, cooktop/oven combinations) require a 40A or a 50A circuit.
The 20A breakers do sound correct for the lighting and receptacle circuits, although you will need more of them, as every dwelling unit must have at least two dedicated 20A small appliance branch circuits and a 20A bathroom receptacle branch circuit; if laundry facilities are present, yet another dedicated 20A branch circuit must be provided for the laundry room outlets. You will want to use a double-pole breaker for HVAC, though: all but the very smallest HVAC appliances require a 240V circuit, and this includes packaged terminal units, even though they only draw a relatively small amount of current (<20A) compared to a conventional air conditioner.
The overall capacity of the subpanel you have on hand (100A) is more than adequate for an apartment-type dwelling unit; however, you will want to check the number of breaker slots available to you in it -- full-width slots are at a premium these days due to AFCI requirements, which makes depending on tandem breakers to fit all your circuits into your panel quite unwise.
Your #2 wire will be adequate for the feeder conductors provided it is copper and of a 75°C rated or better insulation type (such as ordinary THHN/THWN). If you are using aluminum wire, I would upsize to 1AWG -- 2AWG aluminum is marginal for 100A service, requring 90°C rated insulation in order to be at all usable in such an application.
That is an old "rule of six" panel, which while grandfathered, is illegal under its grandfathering becuse it has 7 main breakers. Going to five is a good plan.
It is a classic "CH" panel which is a very good industrial grade panel, except that the 3/4" breaker width make non-ordinary breakers very expensive (a trait it shares with Square D QO). That makes it perfect for what you plan.
On your subpanel which would be near this panel, I would get a panel with a main breaker, with an eye toward (at some point in the future) cutting it over to be the main panel. In a subpanel, the "main breaker" is nothing more than an on/off switch, it is OK for it to be larger than the feeding breaker.
I would also get a rather large panel, at the very least 42 space and even 60 or 84 if practicable: because panel spaces are dirt cheap and often even come with free breakers, whereas running out of space is painfully expensive.
I would aim for an industrial grade panel of good repute (one available in 3-phase variants, not Homeline, BR, or second tier brands) and avoid the expensive 3/4" breakers (not CH or QO).
Over time, as you find it convenient, i'd migrate all your 1-pole and smaller 2-pole circuits over to the new panel.
For your garage panel anything would do, but I'd go for the same type as your indoor panel, so you can use some of those bonus breakers. Again it's false economy to scrimp on spaces, I'd go 20-30 at least.
Also, since garage spaces need to be on GFCI, consider getting a subpanel which has a "main breaker" which is GFCI, that way all the breakers in that panel would be protected (at the cost of potential nuisance trips, a big deal if you keep a freezer in the garage).
Ed Beal raises some very good concerns about overall capacity. One problem with these "rule of six" panels is there is literally no main breaker to stop you from drawing more than 150A. So it pays to be conservative.
It's a difficult situation because you have two big loads that operate sporadically - the EV charger and the range. And the A/C as a wildcard.
One thing I might suggest, is feed the garage subpanel from the new primary subpanel. And then move everything but the range over to the new subpanel. At that point the only things still in the CH panel would be a 60A range breaker and a 100A subpanel breaker. Even at max, those two could not overload the 150A service (by enough to matter). This would force your entire house (from A/C to EV charger) to share 100A, but would remove the possibility of an overload. This would also save you the $85 you'll spend on a second 100A CH breaker.
Best Answer
Upgrade that tiny sub-panel. For under $200 you should be able to get a 30-space Eaton panel that will let you re-use all the breakers in your existing 8-space Eaton panel, plus wire and sub-feed lugs for your Siemens main panel. Speaking of which, that BR240 breaker in your main panel that feeds the pictured sub-panel is an alien - it does not belong in there. It may seem to fit, but bus stabs and breaker jaws have subtly different shapes and mixing incompatible types can lead to arcing and bus damage. If you keep the sub-panel as is, you need to replace that with a QP240.
Note, if you replace your sub-panel with a bigger Eaton unit, you can recycle that BR240 by moving it into the sub-panel and use it to feed the new oven.