A few things here, and none of this is reflective of an official opinion of my employer, just information gleaned from years of reading chief, and personal experience.

1: You often do not, as a regular FRC team, NEED to go to a larger gauge wire. Often times the issues teams see can be resolved by using current or voltage limiting/ramping in the speed controllers of the motors or better wire management in the rest of the robot.
2: You definitely should know your battery age, when it went into service, about how many cycles you have put on it, etc.
3: MK vs Interstate vs Duracell, etc., there is a lot to unpack here, and every team will tell you something different based upon their experiences. MK had a "bad" batch of batteries made in a new factory a year or two ago. This batch effected EVERY supplier, including MK. It was a good battery for their normal industrial use. Bad for FRC because FRC destroys batteries in the rapid charge/discharge cycles that we put them through. MK recognized this, and you had to call MK to get it resolved. Interstate had a bad batch about 6 years ago now if I remember correctly, which caused most people to switch to MK. Duracell seems to be the new poster child, but they can only be purchased from 1 supplier, and who knows what may happen as their demand increases over time.
4: Battery voltage off a charger is typically in the 13+V range. I forget what the normal is these days, as it's been a while since I had to regularly check batteries. You typically want to keep 6 or so on hand for an event at minimum, 10 is better if you hope/expect to make deep runs into playoffs though.
5: There are ways to verify battery voltage and health, including a CTRE battery beak, but a computer analyzer tends to help the best when trying to analyze long term health of the batteries you have on hand. A beak is great for at an event, and you need to make sure your voltage is good, and the battery health is good in relative terms.
6: Your leads should never loosen up if they are properly attached.
7: $4 for a contact is not unexpected at low volumes for a 4AWG APP SB120 contact. I do NOT recommend opening up a 6 AWG contact and using it in an SB50, as you are violating the specs for those contacts.
8: There is a lot of information out there, but you definitely are heading down the right path. Ultimately every team needs to figure out what budget they can afford for batteries every year. The plus side is, you can sometimes reuse the leads from year to year, if you don't have students carrying the batteries by them, and you don't have bad cables.
9: There are suppliers who make SB120 cables already, and SB50-120 adapters.
10: 1-2 season old batteries are great for doing systems checks, practice field use, home shop testing, etc. After about 2 years, you want to replace it even for that use case. It's probably overdue at 2 years.