Am I correct in thinking the Focusrite Scarlett Solo which was recommended would not have that problem?
You need to look at the specs. Behringer seems not to spec the output level, but at least 2 Vrms (equal to +6 dBV) for an unbalanced RCA output or +14dBu for a balanced output on a recording interface product would be "standard". I know, it's confusing that there are several ways to spec signal levels, but the dB concept works better for folks in the recording or live music end of things.
Also, please don't take my comment re the Scarlett Solo as a real recommendation--it was more of a suggestion because it doesn't have as many features that are on the Behringer unit that are not needed for use as a USB DAC/pre driving powered monitors.
If using the UMC204HD, should I use the RCA or main outs, or does it not matter? And whatever the answer is, can the volume of that output be controlled using the dial on the front of the UMC204HD?
Yes, the main output control is what you'd use to control the playback volume.
You can use either output, based on what I read from the Behringer manual. The RCA output would require cables with RCA plugs on the interface end and 1/4" TS plugs on the monitor end, or use of RCA jack-to-TS plug adapters with a set of regular RCA cables. The pictures in the setup guide show 1/4" TS plugs being inserted into the "main" jacks.
If using the Solo instead, I believe the RCA outputs of the Solo are unamplified and therefore suitable for connecting to already-amplified active speakers like this, but I don't yet understand the "balanced/unbalanced" concept, so please let me know if there are any concerns regarding that. The Solo's outs are unbalanced, according to the user manual. Assuming this is suitable, does it matter whether I use the RCA outputs to connect to the monitors' XLR inputs or balanced TRS 1/4" inputs?
Balanced connections are most useful in a high noise/interference environment, like live music, where microphone signals are at low levels and electromagnetic interference from power and lighting cabling would induce hum into long runs of unbalanced cabling. Balanced cabling essentially cancels that interference out by using twisted pairs. There are quite a few explanations on how it works on the web, so I'd just keep Googling and read several as one of them will definitely clear it all up for you.
In the case of most monitors like these, the TRS inputs are designed to take balanced or unbalanced signals. Inserting a TS plug into the TRS connection works just fine, as long as the output stage of the source can swing 2Vrms or so.
