New 2003 Trail-Lite

By law, you need a working battery when pulling the trailer if you have electric brakes. It's essential for powering the emergency break-away switch for your brakes.

Congratulations.
I believe the Trail-lite organization folded a couple of years back.
Black and grey tank monitors hardly ever work. Can't speak for your battery monitor, but you probably should have a battery installed.
I had a 2007 Trail-lite 23QB that we loved until it started to delaminate. It had too much water intrusion that I knew nothing about until it was too late.
All the covers and vents are not trailer specific. They can be bought from most any RV supplier.

Welcome to the forum and congrats on your TT.

If you're very new, there is a lot to learn but, it's fun stuff.
Don't get too concerned about the tank sensors not working properly. They're not known for always being accurate.

I assume you are plugged into shore power?
The battery should be charged and all connections clean and tight before you start to diagnose any electrical problems.

Read The 12 Volt Side of Life, it may help you understand the 12 volt system a little better.

Many of the parts you need can be found on Amazon. Decide what you need and start looking.
Don't hesitate to ask questions on the forum, there is a lot of experience and information available.

I think the first important thing is, check your roof for leaks, rips or possible problems. A small leak can cause a lot of damage if not repaired quickly.

Eternabond and Dicor Self Leveling Lap Sealant are two of the best products for rubber roof maintenance and repair.