Q1) Will the Tiagra shifters work with the RX-100 derailleurs to shift the 7-sp cogset?
No.
Q2) If not, can I just replace the cogset/chain to a 9-sp, or will the RD need to be changed as well?
Yes, and probably yes.
Q3) If I DO go up to 9-sp., do I have to replace the chainrings up front to accomodate the narrower chain? (also, is RX-100 bolt spacing @130mm?)
Most likely, and probably yes.
Q4) Is 8-speed an option that would make any of the above easier to deal with?
No. 8-sp spacing would work, maybe, with your old stuff, but not 9-sp.
Q5) Am I missing anything?
Cold setting your dropout spacing? You need, from what I can figure, a new cain, a 9-speed cogset, 9-sp rings, 9-sp rear derailleur. I can't imagine a bike with rx-100 components is worth it.