Not knowing the exact details of the scope makes it hard to say for sure, but most old scopes are more than enough. First off, the tubes arcing and dying aren't related to the hum likely, but could be an indication that the amp is biased a bit hot. I built an AC30 from hoffman's layout and I had to adjust the bias resistor up a bit, as I recall. it calls for a 50 ohm but that's based on Vox's original designs and they ran them a bit hot. I think I ended up using like a 125 ohm resistor, but I can't recall 100%. Effectively it would be a good idea to go buy multiple values around that range at 10 watts, say 75 ohm, 100 ohm, 125 ohm and 150 ohm. Then take measurements of the current going through the resistor. The process is pretty straight forward. Measure the voltage drop across the resistor, and the resistor value exactly (It may say 50, but a DMM may show 48 or 53 or whatever, get the exact measurement) and then do the Ohms law to calculate the current. You can also just use a simple calculator site like weber's:
http://www.tedweber.com/webervst/tubes1/calcbias.htmthen get the current you're seeing and that's for 4 tubes and the above is for 1 tube. Make sure you're within that tolerance or if you really like the tone now, and don't mind burning up tubes regularly, you can leave it as is.
Ultimately for the hum, it depends on the type. If you can turn the volume up, with no jack inserted and hear the hum, then you can put the scope on the signal pathway and look for where it 'arrives' that would then indicate the trouble location. Is it at the input, or is it at the PI, etc. make sure to measure the signal after the coupling cap or you'll get no reading because the DC will throw a kink into the measurement (I'm still new at scopes, so maybe I've been doing that part wrong, not sure)
~Phil