What is the maximum sub-panel amperage I can use, and is my setup okay?

Deciding the max size of a sub panel is about how much you can load the main panel, and you didn't provide any of that information. So for example you can have a 200A sub panel, but not off of a 100A main panel. Determining the maximum load you can have is NOT counting the breakers and their ratings, you have to know that actual loads connected to them. There are rules in the NEC as to how you must determine this. As to how many 15 and 20A breakers you can have in the garage, that's still a function of the load you can add. You have to start there.

Secondary issue: your #2 Al wire. Your wire is good for 90A max per code (maybe 75A if it is 60C rated), so at best you could use a 90A circuit breaker in the main as your feeder to the sub. But if for example you can only add a 30A load to your main panel, you will have a VERY difficult time getting #2 wire into the terminals of a 30A breaker.

Can I lay the cable direct-burial, or does it need a conduit?

That depends on the specifics of the cable. There are, in general, three types of cable:

• Cable that can be direct buried.
• Cable that can be run through conduit outdoors but can't be direct buried. (To be honest, I am not 100% certain that there are "Big cables like #2 Al, OK for outdoors/wet locations, but not OK for direct burial", but in theory there could be.)
• Cable that can't be used outdoors. This is the case if it is not designed to handle a wet environment, as direct bury or conduit in the ground are both going to get wet. (This includes, at least at the smaller sizes, typical Romex type cable.)

What kind of cable do you have? Upload a picture of the codes printed on the cable and we may be able to figure it out.

What is the maximum amperage breaker I can use at the service side to supply the sub-panel?

That depends on the capacity and type of your panel (can't push out more than you can get into the panel, plus can't overload bus stabs, available breaker types, connection types, etc.) But in theory, #2 Al SE cable is good for up to 90 Amp.

Remember that the subpanel could be a much bigger panel (100 Amp or more). The limit is that the breaker in the main panel to supply the subpanel has to be 90 Amp or smaller.

What number of 15 and 20 amp 120 volt breakers can I use in the sub-panel?

As many as you want. The limit is not based on the number of breakers (e.g., you could have one breaker for 10 receptacles or 10 breakers for 1 receptacle each, code doesn't care except for certain specific requirements such as bathroom & kitchen minimums). The limit is based on expected total simultaneous usage. A little more complicated in some cases, but basically add up everything you may use at the same time:

• Lights
• Power tools
• Electric vehicle charger

If the total on each leg (240 V on both, 120 V on one or the other - try to balance within reason) is < feed breaker size, you're good.

That cable's absolutely worthless to you in this situation...

First off, that cable's worthless for what you need it to do, as NEC 338.12 point 2 says that type SER (and SEU for that matter, not to be confused with USE) cables cannot be run underground, conduit or no conduit:

338.12 Uses Not Permitted.

(A) Service-Entrance Cable. Service-entrance cable (SE) shall not be used under the following conditions or in the following locations:

...

(2) Underground with or without a raceway

...and you're better off just slapping a fat conduit or two in anyway

Besides, starting with the wire is an absolutely backwards way to do things, especially considering that installing fat conduits (a 1.5" or 2" PVC for power, and an additional 1" PVC for telecom if you wish) in your trench means that you won't have to dig things up later if you decide your shed needs MORE POWAH. You'll want to use prefabricated sweeps to bring the conduit up at each end, along with expansion joints to take up ground movement. You'll also get a slight benefit with regards to burial depth here, but it's not enough to be particularly significant, so going down 24" is recommended anyway.

Once you've done that, then you can pull a bundle of individual THWN or XHHW-2 wires through the conduit for your hots, neutral, and ground, getting to pick the gauge of wire based on how many amps your shed plans actually call for and how much load your service can support instead of having to limit them based on the cable you have on hand.

GO BIG OR GO HOME

The other factor is that there really is no limit as to how many breaker spaces you can have on a single feeder, and nobody ever complains about having too many! As a result, I'd grab the biggest panel you can find, space-wise; in fact, I consider a 100A or 125A, 24-space panel the bare minimum for a generic outbuilding situation. Note that you'll want a main breaker panel here so that you don't have to provision a separate disconnect for the shed; it doesn't matter that the main breaker in the subpanel is bigger than the feeder as the feeder breaker in the main panel is protecting everything anyway.