Zap:

can I just plug-my guitar directly into the eclipse at instrument level to use it as some sort of DI

You can do this and it should certainly be better than a sound card. But, there are many factors that influence sound quality, especially with a guitar. Try it and see if you like it – you probably need to adjust the gain (LEVELS {IN GAIN}). The 1/4" input is designed to accept all but the lowest output guitars.

Zap:

what becomes the master digital clock

If you are connecting digitally, everything typically runs at the same sample rate. This means that something somewhere has to determine the sample rate, or master clock. In a simple system, this will be the first thing in the chain – in your case your PodX3. The digital connections will maintain the clock rate down the chain. Be aware that this is a complex topic, and the above just scratches the surface…

Zap:

will the eclipse increase the quality of the signal

Pretty much nothing in the signal chain will increase the quality of a signal – things tend to degrade. But, a good effects unit like an Eclipse (or Axe Fx) can make it sound better – useful, but not quite the same thing. All this is very subjective, so you have to try it and see.