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.
A 125A subpanel being fed by a smaller breaker is absolutely fine. If anything, make sure the panel is big enough in terms of the number of spaces. More spaces are dirt cheap when buying the panel, running out of spaces is expensive. I generally go straight for 200A panels even if I mean to feed it off 60A - I'm a bit of a zealot about spaces.
Your 100A breaker is a wild mismatch for your #8 cable. Not even close.
You have several options depending on your circumstances, all radical.
Change the breaker to 40A
A subpanel is not a service. Wiring to a subpanel gets its ampacity off the same table as branch circuits, the common NEC 310.15(B)(16). Since you must pull out of the 60C table, that puts you at 40A.
Revisit the air conditioner/heater load. I find it astonishing that a window air conditioner could possibly be 6kw and draw the same as a massive whole-house unit, unless it uses resistive heating instead of heat-pump heating. Maybe it can work on a 20A circuit. There is a narrow exception for certain motors, hence ThreePhaseEel's question about your motor nameplate on the A/C unit. You should shoot him a pic of the nameplate.
Change cable
If you really want the circuit to be 100A, then upsize your cable to 1/0 aluminum. (Running large copper such a distance is just a waste of money, and invites dissimilar-metal corrosion since the lugs you connect to will be aluminum.)
While this is the minimum size for 100A, it will also satisfy voltage-drop concerns as its drop will only be about 2%.
Beast of both worlds: 100A on the #8
This is beast mode, and not for the amateur who likes living. Great care must be taken in installation to keep the super-zappy stuff away from loved ones.
If the cost of replacing the cable is prohibitive, and you need most of 100A, then your only remaining option is to "step it up" for transmission with transformers.
The power involved is 24,000 watts. This happens at 240V@100A, 480V@50A, or 600V@40A. 40A is within the current limit of your wire.
So you get two 25,000 VA transformers that have 600V on the primary (@41 amps) and 120/240 on the secondary (@104 amps). These things are not cheap, but can sometimes be had used. Canada's 575V is close enough IMO. Transformers are bidirectional, so the house's 100A breaker goes to the 240V secondary of one transformer. This steps up to 600V on the primary, which is attached to the long cable. The other end of the cable goes to the other transformer's 600V primary. Its secondary gives 120/240V.
The transformer isolates, so it must be treated as its own separate service, and the neutral must be bonded to a local grounding rod and the grounding system, since the subpanel now counts as a main panel. Now you can use the service feeder table for sizing your wires (#2 aluminum) between transformer and now-main panel.
Of the works, only a tiny bit is 600V, and that is isolated, so a person has a chance to survive casual touch. Still, the installation should be designed to minimize exposure, e.g. Have the wire go straight to the transformer, no splices or reroutes, and put the transformer in an enclosure or outside.
Best Answer
Either it's 120V subpanel and he's plain paralleling... which is rather bad.
Or it's a 240V panel, and he's supplying hot/neutral/ground with one cable and the other hot with the other cable. That's a different kind of bad.
Both are unacceptable. You should replace both cables with a single /3 cable:
And, both hot wires need to land on a 2-pole breaker which has common trip, meaning both legs trip together, and, move either handle, the other goes too.
Note that a duplex/tandem/twin/double-stuff has independent handles and must not be used, as it will create a dangerous situation if you do.