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Dave I just watched your War thunder stream, and had a couple helpful hints.
In arcade battles, always be WEPping. Always. There's no drawback. In realistic/simulator you can burn out your engine, but in Arcade mode WEP just turns off after about a minute until the engine "cools off" then re-engages again.
"crews" at the bottom of the screen is just a slot for more planes. You can have more than 3 planes per battle, you just have to have crews to pilot them (many of the planes carry more than just a pilot). So basically those are "plane slots." You can open the first one up there for 10k, and the second one for a similar amount, after that it takes golden eagles (premium currency) to get more crews. You've already been awarded one plane, Thatch's F2A Buffalo, so you should get a crew and then assign the plane to that crew.
As you play, you get research points for shooting stuff/bombing stuff. These research points go to both upgrades for the plane that gets the points, and to work toward unlocking more planes. That big "research" panel you opened at the start is the list of things you can research.
"I feel like I didn't do much" you killed 7 ground targets. That's a very non-trivial contribution toward victory (victory in this game is almost always based on killing ground targets/bombing bases). Shooting down other planes is fun, but it's rare (especially in arcade mode) that there's ever a victory come from shooting down planes.
"Assists" just means you did damage to something and then someone else killed it. You still get points for that (research points and silver lion currency).
Your machine guns have a maximum effective range of about 800 meters (2500 feet). Hit the button (I think it's button 3 on the mouse, or maybe 2, I forget) to put a targeting bracket on what you want to kill, it will tell you the range. wait until it gets to 800 meters to start shooting. It's possible to hit something further off, but you have to really allow for gravity. If it's not standing still, forget it.
Also, you definitely want to put targeting brackets on any plane you chase, as that will make it easier to follow if it flies outside of your field of vision. It's rare that a dogfight doesn't turn into a furball, especially in low tiers, so it's important to keep track of your quarry like this. You can also hold left ALT to make the camera track your current target, very useful.