Avoiding missile damage early game

While a system scanner might indicate to you whether or not a ship is present in system, there's no reliable way to avoid enemies equipped with missiles without avoiding enemies entirely. Some are just going to have them. Automated ships (with AI rather than crew) tend to be the biggest offenders (and, annoyingly, also can't be captured intact, making them good candidates for fleeing if you don't feel you can kill them quickly).

About the best thing you can do (since you ruled anti-missile drones out) is to just be sure to target their weapon systems first thing off and keep pounding them so they stay down. That's just good advice in general, regardless of the enemy's weapons, but critical when dealing with enemies who fire missiles. It's probably even worth one or two of your own missiles when targeting these systems to make sure you get them down ASAP.

If even that is causing you trouble (as per your "I can't hit them fast enough to shut them down" comment), you're kind of screwed. I mean, some missiles are just going to get through and you have to deal with it. At that point, the best you can do is get good at recognizing which enemies are going to be more trouble than it's worth to kill, and flee them early. Honestly, though, early game you really shouldn't have to flee much, at least in the Kestrel. Just make sure you save your missiles for the heavy hitters instead of wasting them on ships that can't do much to you anyway.