Cynical's answered the question well, though i'd like to chime in and warn against using Mage Armor as your staff spell. Since it's always going to be level 1, you won't be getting any benefit from it once you've runed up your robes to a basic level of enhancement. Plus, since it's a spell that lasts all day, there's little benefit in putting it in a staff to begin with, as the main benefit of being able to "spam" the spells inside of it would be wasted.

For the question about cantrips, nope! The extra cantrip is yours to cast freely, with or without charges remaining on the staff.

For the note on changing spells I don't think you can normally through the feature itself, though you could retrain into the same feature with different selections down the line if needed.

For your last point, since you can technically retrain by taking some of your downtime to basically roleplay as tweaking/re-carving runes on your staff, I don't know if this is as much a worry for you. You can get good mileage out of things like magic missile as a fallback damage spell, true strike is a also great for trading higher level spell slots that you may not have as much use for at the time for getting your hits to land if you pick up any attack roll spells.

For level 1 spells on the staff that aren't True Strike/Magic Missile, I've always liked: Befuddle, Gust of Wind, Grease, Shockwave, Command, Mud Pit as these mostly stay useful with using those level 1 spell slots and extra charges you can get. Mostly depends on what flavor you're going for and potentially what staff you want to craft, or are able to get to upgrade your makeshift staff into.

I'm not incredibly experienced, but for the first question, AoN cites Core Rulebook pg. 592 as:

If a staff contains a cantrip, you can cast that cantrip using the staff without expending any charges. The cantrip’s level is heightened to the same level as cantrips you cast. So I don't believe a cantrip requires charges to cast, even from a Staff Nexus staff.

And I'm also not seeing any contrary guidance on the unchangeability of the Staff Nexus spells, which quite frankly is a little odd to me, given the rest of the system's more flexible approach.