So, there's a thing characters could do where they make custom missions. I have a simple question that, hopefully, will have a simple answer.

Should I make a mission that has a personal overtone, am I allowed to make missions that function outside of the village? What I mean is, can I make a mission that in no way involves the Kages, that works outside the standard operating procedure or even outside of the law in general? Obviously, rewards for missions like this are basically common sense and would usually result in gains lower than when they go through the proper channels.

The reason I ask is, it's more than possible that as I gain a bit of traction in the world, I might be tempted to actually go on adventures that, shall we say, aren't part of the normal ninja routine. Some of these adventures might very well lead to financial rewards, and I wanted to know if I could make those rewards tangible, as well as set up a solid framing device for these escapades.

I can see that being possible, but not as a Genin. Genin are basically right above academy level, so going out on self-adventures that grant rewards is a bit out of the range of a Genin.

HOWEVER. It may still be possible if you had a mentor who went with you on those adventures, such as a leave of training, like what Jaraiya did with Naruto in canon.