Gameplay is easy: Sufficiently detailed 'Mechs, them being distinct; and them being affected differently in combat. In other words, OP's points 1 and 3.
I don't care for construction rules in themselves, i don't play with custom units despite designing them often. That said, given that combat rules demand certain level of standardization, construction rules work easily with that, despite some logical issues (such as all 'Mechs having exactly same space in all hit locations).
As for the universe... BattleTech offers a lot of possibilities. There is little i don't find acceptable, though i will note that while i don't necessarily find even acceptable things of any interest.
Some core rules are:
-No aliens. Non-human aliens tend to be unconvincing and in case of well done aliens, the thing becomes about them more so than otherwise. As it is, the Clans offer an excellent alternative by having humans with frankly alien society that works (often fiction refuses to even contemplate human societies that aren't fundamentally somehow different).
-Hard science fiction aesthetic, with strong emphasis on aesthetic. BattleTech is not hard science fiction, but it pays lip service to it. There is no artificial gravity, no energy shields, all energy weapons are based more or less on real principles (no "phasers" or "blasters"), etc. BT's few overt violations of hard science fiction are its magical fusion thrusters (really, they're pretty excessive) and its FTL technology, both of which can be excused as being sine qua non of space opera.
-Keep things on the ground. While BT is space opera, it is unusual in the sense that its focus is on planets much more so than space. Space is secondary, it may make for interesting battlefield but that's all. Indeed, as it is, i feel existing WarShips are almost a violation of this, and i wouldn't care if they were excised from canon completely.
Not quite a core rule but strong preference:
-Focus on interstellar empires, not megacorporations or minor factions. Megacorps in BT work as a minor part of the setting but i don't care to see them in roles like they do have in many modern scifi works or how they typically are in (post-)cyberpunk. They are part of the system, not the system. As for minor factions, BT is a space opera, so focus on big players. Major Periphery realms are about the smallest i think is OK, other things can exist in fiction but i really don't give a shit about them.
Cache makes a good observation with how Jordan Weismann starts BT. Those are indeed good principles, at in the event of a time jump or whatever, i think those aspects should be reinforced or focused on to an extent.
I would modify "tech degradation" though, i feel BT has place for highly advanced tech nowadays, like cyborgs or advanced weapons. But "tech degradation" can be portrayed differently, eg BT could have high tech warfare but note how it affects the world, what it requires, what its cost is.
I'll note that MechWarrior Dark Age began with "tech degradation" but it was rather different from how it was when BT began. Instead of tech being rare, reduced armies meant that improvised weapons were used more, and theme was a reliance on communications and how them breaking down affected things. As such, a new BT era or whatever could play the "tech degradation" angle differently rather than focusing on LosTech.
