So essentially, just a temporary triathlete?
I don't think they should have infinite skills. Both the level designer and the player have to count up how many skills each [however long the invincibility lasts for] second section would use, and then figure out where you need to utilize them to finish with enough skills. That just seems like it would get annoying, and it would be much less hassle to have the actual number of skills you had for that section, to see what you need to do. The only case I can think of where infinite skills would be a good idea is "X of everything" levels, where the player likes to solve them with as few skills as possible, but that's very niche.
If it does have infinite skills, maybe it can still only be assigned levels that are actually on the skill panel as part of the level's skillset. This would keep the original integrity of the level's design at least a bit, but again, I still think it's a bad idea to add at all.
Maybe to add a bit of variety, invincible lemmings shouldn't be able to blow up or become stoners. If you think about it, in reality, it wouldn't work unless the lemming actually died. It would add a bit of variety, make the skill less overpowered, and create more puzzle opportunities.
100% no on destroying steel. Steel should be indestructible above all else; that's the whole point of steel.
but, then I realised - what if they're assigned a Timebomber?
I would simply add the number above it. Maybe (and this is in general, not just for the invincible lemming), the invincibility timer could be blinking gold, and the bomber timer could gradually go from yellow to red for aesthetic purposes and to tell which one is which.
I think it should be a lemming type that can be pre-assigned and assigned to a hatch to maintain consistency with the theme of the other lemming types. However, I wouldn't be against it being a skill either.
I don't see any reason why they should walk past the exit.