Since I've got someone at my table currently building an AI, I've checked on the rule to help him, and there are a few things that I can say :
-If you want to use the Anthrodrone as a home device, you indeed should get the pilot origins quality. If you are going to remote control it from your classic home device (be it a deck or a rcc), then you do not need it. Considering that you are probably starting as a "normal runner" (so 25 + negative qualities karma), I'd say you should take it only as rank 1 for now if you want it, and upgrade it later at double the cost.
-Depth is kinda tricky. I know people who say it is worthless, but after checking into it, i don't really agree. Depth plays in way too many things for it to be forgotten : it plays for your Core monitor condition, the number of programs you can learn, and the maximum attribute you can emulate is equal to your depth. And it's also your edge. If you are going to hack using emulate, you definitely need a decent Depth.
-Cybercombat really sucks for AI as far as I can see, although some of their advanced programs can probably give them a good edge if well used - I haven't check into those a lot. Either way, unless you are confident in your defensive capabilities, stealth is your way to go in the Matrix.
-If you were on the grid or an other device, you could still remote control the drone. But from a host, I don't think it would work, since you can't interact with devices on the grid from there.
-There is apparently a program called Deicide, mentionned only once in the Data Trails, that allows one to deal Core Damage to an AI loaded on a device the same way the Biofeedback program would on a physical user. You will usually take less damages than the device, but sometimes you aren't lucky with rolls, so there is the possibility that you get disrupted in this way while your device is still running.
As for Overflow, one interpretation is that when a device gets bricked, all the programs running on it get shutdown/deleted, and that's what is happening to the AI : the way the overflow works is a way to interpret how the AI must struggle to avoid deletion when its home device no longer works.
-It seems that you have to pay autosofts with karma
I haven't answered everything, but hopefully that will help you a little bit.