Did any UN Units get really really Lost in the Chaos from the War and Environmental collapse?
Quite a few UN-inserted special forces units and black ops went completely radio silent well before the Unification War. The UN kinda shrugged and was like "yeah either they're dead or rogue, maybe one of them did New Gorbals? We dunno." Actual infantry and mech units were much more well-tracked, were fighting in a relatively contained area, and had much less autonomy, so not a lot of them went missing, though you're always going to have a few MIA soldiers, just because their bodies weren't found or they got captured or whatever.
Can I give the Tharsis Region some Love as a Bebop Fan?
I'm afraid that in terms of Martian geography, I am no expert, and thus I am using an alternate system of naming regions based on the original game that this quest is based on. So, I have no idea where the Tharsis region may fall in terms of this universe.
And Three, Can I basically make them the UNSC Marines in terms of basic equipment, sans more effective armor and helmets to protect them from the enviorment, as that's about the only image I can get for them?
To talk about UN equipment and military doctrine briefly, though please forgive me for any terminology mistakes, I am by no means an expert - UN standard equipment tends to be pretty high-grade. In addition to the mechs, you've got power armor, you've got infantry weapons and armor that basically matches military-grade pre-Collapse weaponry.
In that vein, pre-Collapse doctrine on warfare tends to be "limited ground engagements by power armor backed by heavy orbital/aerial support, if aerial/orbital support is unavailable for whatever reason, utilize power armor to deal a decisive defeat supported by infantry/local auxiliaries as needed", which the UN seems to have adapted. The average UN ground soldier is going to be either a power-armored guy, specially trained in being the vanguard of combat, acting as the meat of the engagement, or an infantry soldier trained in operating as support for that group of power armor/mechs. That infantry soldier is going to typically be specialized if they aren't in power armor, because for the UN, power armor
is the primary force that they want their troops trained in.
I am going to warn you, though - Omakes regarding the UN are highly unlikely to be considered canonical, due to the many unknowns surrounding them.