Trying to plan an out of state trip and Missouri is in my scope. I'm curious what to expect for beaver and otter populations, is it mostly river trapping by boat? Is there ROW trapping?
When do things ice up and when do things thaw?

Nearly all land north of the Mo river is flat private farmland, but river trapping should be available, tho logistics wll be a problem.
A huge % of land south is National Forest, and open to use. Maps can be had from the USFS, AND USGS. I would recommend waiting till after Mo gun deer season. Float trapping rivers can be great fun and productive, need to plan logistics in advance, ie: put in's take out's, camping spots and shuttling boats. Plenty of river outfitters would be available to assist with all this in the off season. PLAN AHEAD!!
Don't forget a non resident permit. Mo is a gun friendly state, constitutional open and CCW carry.
Plenty of Beaver, Otter, Mink and Rats, as well as land animals too.

www.evertrap.proboards.com is a mostly Mo trapping site, with a great bunch of guy's/gal's to help with ?'s.

big bob

Is there ROW trapping? How much do the rivers tend to fluctuate?

The Ozarks tend to be steep and rocky, any kind of rainfall runs off down hill to the creeks and they will come up and go back down very quickly.
If by ROW trapping you mean accessing rivers then yes, sort of. Most bridges and crossings will have an access, frankly I'm not sure about bank/river bottom property rights. You have a right to float "any commonly navigable waterway". You need to research specific sites.

The guy's at www.evertrap.proboards.com will know better.

I've done a little out of state trapping in the Ozarks not for beaver & otter tho I was mainly running cable restraints for yotes. However when it comes to public roadways the Missouri Regs book does say

"You may not take any wildlife from or across a public roadway
with a firearm, bow, or crossbow. A Conibear®-type trap may
be used adjacent to public roadways only if set under water in
permanent waters"

Missouri Regulations

The best help I had when I would trap there is I would call the regional game & fish office and ask to speak with a game warden they were always happy to answer any questions I had and offer advice for out of state trappers

You might want to check on the drought that is ongoing in a lot of the state. You're liable to be pulling your canoe/boat over more gravel bars than paddling in water.