Syntho, wrong forum!
two power tubes are creating a bit of noise when tapped. The other two power tubes don't make a sound, and neither do any of the 12ax7s.
This is normal. Tubes differ in their internal construction. Some are more microphonic than the others. (The old Mullards in my Marantz 8b ring a lot).
..plastic mesh fabric near the top vents on the amp that started to melt.
This is indication that output tubes got too hot and, most probably, their operating point was/is wrongly set.
Without schematics, without knowing what amp, what brand of tubes, what plate voltage, what operating conditions, I can't say anything more. If it's one of the chinese creations - good luck with it, it could be anything. From wrong resistor values to leaking coupling caps to incorrect operating voltages..
Long ago I remember having my amp turned on without my cab being connected.
Bad idea. You were risking output transformer damage.
I also don't have the $$$ for both a retube and to get a tech to look it over.
Well, if you dont have the funds to keep your tube amp in shape, stay away from tubed units. Or, stop buying all other junk and invest the money in your amp, if it is important to you. Bring it to the qualified tech and pay him. (Techs have to feed their families too, btw.)
Or, if you have the measuring equipment (scope, oscillator, voltmeter etc.) and are prepared to deal with high (and potentially lethal) voltages, then do it yourself. The Tube Amp Book by Aspen Pittman is a good starting point.