1. Both. However blowouts are rarer than the slow leak. Usually it's a shard of glass, wire from truck tire retread, or something like a thorn causing a slow leak.

2. Riding on a flat will slow you down so much that walking is faster especially on the road. Handling is horrible. Plus you'll ruin the tire and maybe even the rim.

3. Patching is somewhat of an art. The biggest mistake is not waiting enough time (about 5 minutes) between applying the glue to the tube and putting on the patch. You'll also need a pump, tire levers to remove the tire; and possible a 'boot' to cover over the hole in the tire. Another big mistake is failing to find what caused the flat, such as a thorn, and removing it.

There are some other options. One is to bring a spare tube and just slap it in (after finding the thorn or whatever). That will save some time and you can patch the tube later. You can also get some 'scab' patches that don't require waiting for the glue to dry or even applying any glue. Some say that this is a temporary measure and you should still fix the tube with a proper patch later. Others say this is BS. You can also speed up things by using a CO2 cartridge to inflate the tires. I prefer a pump myself as I've had multiple flats on a ride a few times. That's also why I carry a spare tube and a patch kit.

The above comments are not for tubeless tire systems. That's another discussion. Plus I'm about 99.9% sure that your Giant doesn't have tubeless tires.

You are probably correct that your new bike's tires will probably flat more often than the fatter MTB tires. When the OEM tires wear out, you might want to consider getting something like Gatorskins or Armadillos for replacement as they are more puncture resistant.

The first thing that you should do is practice, practice, practice patching a flat in the comfort of your home or garage. Also make sure that you have all the tools needed to patch a flat on your bike when riding. Tip: Those little tubes of glue are known to harden with time especially once opened. It's really sad when you've used up your spare tube and need to use the kit only to find the glue is bad. That's why I check it about once a month by squeezing the tube and also carry a couple of the 'scab' patches just in case.