Electrical – Why does the garage still have power when the house main is shut off

There should be a snap to it; the contacts are spring-loaded so that they make or break contact quickly, reducing any arcing that might occur as they approach or separate. Arcing can still happen, but there's less opportunity for it than if the contacts were moving at the speed that you're moving the switch.

Like any mechanical device, they can wear out over time. And this is something that I'd definitely have a licensed electrician look at; you may even need to have the electric utility involved too since they may want to disconnect your house completely before working on the switch.

Here's what Wikipedia has to say about them.

tl;dr - if you are going to all the work, and a subpanel, you presumably want a bit more than 20 amps (think it needs to be 30 amps minimum for code these days, and 60 amps is probably better.)

You'll have to dig a ditch. At that point, my opinionated opinion is that you should go ahead and put in conduit, and an additional conduit for any current or future possibility that you might want cable, phone, network, etc out there. Ditches are expensive and a lot of work - conduit is cheap, once you have the ditch...Often cheaper (with wire) than "direct burial" cable, and far more resistant to damage in the future - plus it does offer you the potential of pulling out the wire and pulling in new wire if there ever was a problem - but that's low odds of you making any use of it on he electrical side. Network, quite possible.

Before digging a ditch in a city (especially) backyard, call Dig-Safe and have all (Offically known about) services (gas, phone, electric, water sewer & things you may not know about) located. Turn off the circuit to the garage - I would not worry too much about where it is (unless it runs in a conduit that you might be able to re-use - which is not too likely), but odds are that you'll find it when digging, and it's less exciting if it's turned off when you do.

Any wire used must be rated for wet locations - not difficult, just be sure it is. Any exterior conduit is assumed by code to be wet (and that's generally true.)

If the portion of the backyard you are crossing is not travelled by cars and trucks (not crossing the driveway) depth is sufficient if the TOP of the conduit is 18" below finished grade. Be sure to lay "buried electric line below" tape in the top 6" of the trench fill. If you are not digging below frost line (4 feet or more where you are, probably) you definitely need to bring the ends of the conduit up vertically at buildings, and provide a slip (expansion) joint, as the conduit will move with frost. That's generally needed even if the conduit is buried below frost-line as well, unless it's going straight into a basement below frost-line, but its especially critical when the conduit is above frost line. If you don't find the cost phohibitive, a layer of XPS foam over the top of the conduit provides one more indication that there is something there (when someone else is digging, later) and can reduce frost movement a little bit (or a lot if it's wide.) Alternatively, 2" of concrete over the conduit provides some serious protection on top of the conduit, and reduces the required depth to 6" in Rigid or IMC metallic conduit (which may be well worth it in your situation to save on digging) or 12" in PVC conduit.

If you happen to want a walkway that would happen to run where the electric service would, a 4" thick concrete slab extending 6" beyond the conduit reduces the required burial depth to 4" (ie, right under the slab.)

Look for NEC table 300.5 for more detail.