Yes, channels have independent inputs, you can simply wire one input jack to multiple channels. It's a quite common design.
The easiest way to channel switch in that case is to use a switch to mute the channel you don't want, by switching the signal to ground somewhere before the channels are mixed together.
Running the signal to a footswitch and back to the amp can certainly introduce noise. Tube amp signals are typically high impedance and sensitive to interference. This was a fairly common practice in amps in the 60s and 70s though, and is still common for the reverb switching in many amps.
An electronically controlled switch is a better alternative. There are a lot of different options. Relays, Vactrols (resistive optocouplers or LED/LDRs) and Jfets are probably the most common. Each type of switch has its pros and cons. Jfets are cheap and require almost no current, but can't take high voltages. Relays need quite a bit of current, Vactrols are expensive, etc.
A relay is probably the simplest solution. They also have the advantage that you can get a DPDT switch in a small convenient package. You need to generate a supply voltage to power them though. Could be taken from an unused transformer winding, the heater winding, or adding a separate transformer.
For a really in-depth description of switching methods for tube amps, I recommend Kevin O'Connor's book "The Ultimate Tone".