Expansion tank parts questions

I bought a kit very similar to this to replace my failing ex tank:









A couple questions. First, is the black sleeve over the coolant sensor supposed to stay on? My old sensor was exposed, but it doesn't look like the sleeve easily comes off.



Also, what is that oil cooler thing for? I have a manual tranny and couldn't figure out if I just didn't have it on my car or I was blind.





Thanks!