Sorry to hear about the water issues. As was said, you just need to do get those fixed.
As you have discovered, the system is generally PEX tubing. However, if you look under your sink you will see small grey tubing going to the faucet set up. Those are generally threaded and just hand tightened. The PEX connections have been made with a crimp tool. You will need to buy on if you plan on repairing the factory work. Sharkbite also makes PEX connectors which work well also.
As far as the water fill leak, that most likely is a hose clamp type situation.
2006 HW256 (previous pup)
2013 Chevy Tahoe
Equalizer WDH 1000#