What makes it hard in answering your questions is that I'm not sure what you have wired up and to what you don't have wired up. I can try and guess by your descriptions ,but I'm guessing. The trigger wire you refer to from the accessory wire would be used to turn the relays on and off. Do you have power wired in to the relays that'll actually power your devices? Again, the trigger is only for operating/energizing the electrical coils on the relays . Those relay coils when energized, will cause an electromagnetic field to generate, closing the relay contact points. Think of the relay coils as your finger on a light switch. All the coils are doing is turning on or off the switch. You do have to have the switch wired with power to be switched on to the amp, lights etc....

You may know all this, I'm not sure.
The wire nuts should be able to be reused. Depends I suppose on how cheaply they are made. Heat shrink comes in many different sizes and you want to use the smallest size that'll fit. Heat guns are the best to use for heat shrink, but a well skilled lighter will do pretty good.

As far as running the wires under your tank, that depends on the wire and your skill at fishing it through. I like to have
my wires protected with an outside jacket of some kind. If your wires have a thick enough jacket maybe ok. Keep in mind heat and abrasion concerns. Not a lot of real high temp stuff to worry about under the tank.
I have multi meters at my disposal , and if you're in the habit of doing any electrical work, an inexpensive meter would be a worthwhile investment.
If you are indeed getting power to your amp, then are your speakers hooked up right? Did you have a source,ie music
properly plugged into and playing when you were having no sound?