Also from a UK perspective:
screw up
I don't regard this as at all offensive, merely informal.
Chambers (http://www.chambersharrap.co.uk/) has:
screw-up noun, slang 1 a disastrous occurrence or failure. 2 a person who has messed up (their life, etc).
I would say that the first meaning is much more common, either as a noun or a verb, although I think "disastrous" is too strong (see ODO below.)
Oxford Dictionaries (ODO) (http://oxforddictionaries.com/definition/english/screw?q=screw+up#screw__54) has:
screw up
1. (of the muscles of one’s face or around one’s eyes) contract, typically so as to express emotion or because of bright light:
- his freckled face screwed up with childish annoyance
2. informal, chiefly North American completely mismanage or mishandle a situation:
- I’m sorry, Susan, I screwed up
Meaning 1. is standard English - neither slang nor offensive.
Meaning 2. is the same as Chamber's meaning 1. but I would say (1) it's now quite common in British English; (2) "completely mismanage" is a better description than "disastrous".
I would regard this usage as informal rather than slang, and certainly in no way offensive. I do not think of it as related to the offensive slang to screw for copulation.
Also have a look at the ODO link above for screw someone up and screw something up.
to screw
In the sense talked about here, I would regard this as vulgar and offensive.
Chambers (http://www.chambersharrap.co.uk/) has:
screw (verb)
5 tr & intr, coarse slang to have sexual intercourse with someone.
ODO (http://oxforddictionaries.com/definition/english/screw) has:
screw (verb)
3 [with object] vulgar slang have sexual intercourse with.
- [no object] (of a couple) have sexual intercourse.
But note that it also has:
be screwed
be in serious trouble:
- if you’re colour-blind, you’re screwed
Again, it would be worth looking at the ODO link thoroughly because there are lots of ways in which various phrases involving screw can be used -- some vulgar, some merely slang or informal, and some as standard English!
crap
Chambers (http://www.chambersharrap.co.uk/) has:
crap
coarse slang noun 1 faeces. 2 nonsense. 3 rubbish. verb (crapped, crapping) intrans to defecate. crappy adj (crappier, crappiest) rubbish; inferior.
Personally, I would regard its use for defecating as very coarse & vulgar, but its other uses as only slightly coarse - but that can depend on context and tone of voice.
Also have a look at the ODO (http://oxforddictionaries.com/definition/english/crap?q=crap) - the entry is a bit long to reproduce here.