Yes if you use zStart.
There is even an option "Compile, launch and come back to the editor".

If you mean compile just after quitting from the editor, then no; Axe can't do that on its own.
As mentioned Zstart can do that sort of thing, tho.

Just because I'm curios and this is quite odd, but is it possible to edit the bits of a variable using the Pxl-Command on arbitary buffers?
For example we want A to be 10000100, can we do
0->A
Pxl-On(0,0,°A)
Pxl-On(5,0,°A)

I know this isn't very efficient, but can you do that?

Yes, I believe that should work, that's a cool application for pixel buffering  Keep in mind that numbers are stored little endian though and so the "pixel coordinates" for X in 0-7 are actually the low byte and 8-15 is the high byte.

Ok, thanks. I came up with this idea planning a maze generator were setting bits of variables is necessary, but I just go with adding numbers each time.

Use eulers 'e' to access individual bits. You can also use bit masking combined with shifting left/right (* or / by 2) to read/write bits.

You can only use e to read bytes, not write.  His method was a way to set, reset, and toggle individual bits in an intuitive way that didn't require masking

Also relating to absorbing appvars, this is kind of an obscure feature, but it would be nice to be able to absorb only certain sections of appvars. Like say [appvAAPPVAR]<<10<<100 to absorb bytes 10 through 100 of appvAAPPVAR

Pic automatically do that, you just have to change the size of the pic.  Pictures are not limited to heights of 95 and 96, but can actually have any height.  If you send a picture with a smaller or larger height, Axe will import it correctly.  For appvars, I don't think that is really very useful.  If you have an appvar, you probably made it yourself/on a computer.  Why would it be formatted such that you only need part of it?  You could just have 2 appvars, one with the part you want to import, the other without it.