In addition to the Rema patches, put a new tire tube in your kit. The best method for dealing with a flat roadside is to swap out the tube and save the task of patching for later. If you’re not sure what size your tires are, it’s printed on the sidewall of the tire. This inner-tube buying guide has some photos of where to look. What brand is almost irrelevant as many tubes get manufactured in the same place, so whatever your local shop has behind the counter for under $10 will work fine. The only reason to spend more would be to save weight.
It’s both surprising, and not, if you think about it. The way I’ve usually seen it happen is first the rear tire goes flat, then the front one. My theory is that both tires went through the same pile of glass/thorns/metal bits and they both picked up junk. the reason for the flat occurring on the back first is because there’s more weight on that tire, so the offending article works its way through the tread more quickly.
Yes, bicycle shops generally provide repair services. A typical list of offered services includes regular service/adjustment (including safety check; and  brake, drivetrain, and tire inspection); premium, and annual service, which is a more extensive tune-up. More specific repairs include drivetrain tune-up, brake system tune-up, flat tire fixes, brake repairs, cleaning and touching up the bike frame.

I own the Lezyne CNC Chain Rod, which I’m a big fan of (yes, I know I’m partial to Lezyne here–they’re a local-to-me, San Luis Obispo company and they sponsored my collegiate team, so I got tons of gear) and it costs right around $30. But the truth is that even the cheapest of chain whips will likely do the job. In fact, I’ve found a dude who knows his stuff and shows you how to make this tool all by yourself. I would even go so far as to say that you probably have enough materials laying about your garage (if not, try your parents’ garage ? ) to put this together without spending anything. Check out the video below:

Good bike repair stand. I can't say "great," because it arrived with some small plastic pieces broken. These are small pieces that go between the legs and the supports for the legs to act as a sort of buffer to prevent metal to metal scraping. They are just too fragile, it seems. 3 out of 4 legs had these broken upon arrival. However, these are really only "nice" to haves, not need to haves, and they don't really detract from using the stand in any way. Once you have the stand set up, you don't worry about these at all. The main clamp holds the bike well, if you take care to balance the bike properly. The bike may tilt in the stand if you don't balance it - there is a weight limit to the clamp. I like the included tool tray, but I wish it was more adjustable in height (it is essentially fixed in one position). And the tray could be larger. Otherwise, I am quite happy with this stand. I have worked on several bikes using it, and assembled one out of the box. It makes the job of doing things like adjusting brakes and indexing derailleurs so much easier. The price of this stand is excellent considering the quality.
Broadly speaking there are two major wheel sizes – 700c for road bikes and 26 inch off-road wheels. However there are a whole variety of older, less common wheel sizes and of course fold up bikes have much smaller wheels altogether. Repairing a puncture is not an emergency! It’s a basic procedure that every cyclist should be able to to perform, but doing it correctly involves more steps than many people realise. Wheel building is often seen as the most artful skill in bicycle maintenance, and separates the professionals from the amateurs but that doesn’t mean you should consider it out of reach. Truing a slightly misaligned wheel is a cycle maintenance job that almost any cyclist can perform.