Can I mix these SRAM Apex components with Shimano/Ofmega components on my bike temporarily?
Proof's in the pudding, but generally you should be fine with this setup as long as the frame's OLD is large enough to take your new rear wheel hub.

The trick is to keep the front mech and shifter compatible with each other, and keep the rear mech and shifter and cassette compatible.

One interaction can be that the narrower width of the 10 speed chain takes longer to move chainrings at the front because the front mech's cage is for a wider chain.  For a beater bike you might squash the cage horizontally a bit, or learn to live with it.

Also the RSX shifter on the left of your bars will be mostly grey, and the new one on the right will be much more modern-looking.  A workaround here is to use a downtube shifter on the left for the front mech, and leave the left brifter as only a brake.  This was a thing in the 90s, noticeably some yellow texan.