Hi David,
With most SS amps you should be safe letting one channel run without any load connected, but you have to ask the manufacturer first. If it is a famous brand which has been around for a while it is probably OK. Only a fool would sell an amp that is unstable with no load and he won't last long. But a diy or custom amp should be tested on a scope for stability first. In any case, rewad the manual and if you smell smoke, shut it off.
With most tube amps you MUST connect a load to the outputs to prevent damage to the output transformer. When there is no signal coming into the amp it is OK to not have a load connected for a short time to change wires, etc, but the manual will always say no.
In most cases you can split the line level signal to drive two amplifier inputs. The exception is a tube source with high output impedance cannot drive two solid state inputs with lowish input impedance in parallel, you would lose some low bass but it wouldn't hurt anything. But in most cases you
cannot split a speaker level signal to power two separate speakers connected in parallel. The load would be too low impedance and few amps can handle that. Speakers and amps could be damaged.
So what you propose is a good solution technically, it should sound better. It's called bi-amping. But if you want stereo you need two stereo amps and 4 speaker wires.
If you do it in stereo, the two amps left and right need to be the same amp, or at least very similar. Not tube one side and SS on the other side.
Have fun!!!
Rich
