1. If you're doing the break-in online, only the laps of a Race count towards adding mileage to the car. Free Run laps do not seem to. I do not know if there's a way to make them count or not. If there is, I haven't found it. You can do Free Run Laps all day on a car with 0 miles and it will still only have 0 miles on it when you go back to your garage to check.
I sometimes do break-in's online, just to have some company, but often times, there aren't many people around to run with, or a bunch of noisy people singing poorly in the mics. Sometimes you get to hear people arguing with someone in their house, too. However, there are also times when you jump into a fun room with cool, funny people. That's when it's worth it.
2. 300 km is the equivalent of
187 (rounding up) miles. 300 km/187 miles seems to be that magic number where it gets the most out of the engine, but without damaging it.
If you're doing your break-ins on SSRX, then you need to do 10 laps. In practice mode, the Free Run laps do count, which makes perfect logical sense to PD, apparently, but not to anyone else.

You can change your engine's top speed in the transmission settings to get a higher max speed on SSRX. It'll save you some time.
If you're online, and the host hasn't
foolishly turned off the boost/draft, then that will serve to allow you to go even faster with the slipstreaming. Even slower cars can get over 200 mph. If you're in a good room, people will drive in a cluster drafting each other making it faster to complete 10 laps.
3. Some other races that people use to hit that many miles are the shorter Endurance races in A-Spec or B-Spec (some people prefer not having to drive the laps themselves and let the B-Spec "Bob's" do it for them). Grand Valley 300 km, Tsukuba Roadster, and Indy 500 are often mentioned, but you should most likely quit Indy just after 75 laps. Yes, to answer the follow-up question, the laps that you do before you quit, do actually count.
Since some people get confused; what that means is that it will say 76 on the lap counter, not 75. You'll want to
complete that 75th lap, not just get up to it. You can also check your mileage in the garage to make sure that you've indeed done enough. Once you have, go for the oil change and you'll see that you've gained power over what you started with.