Are "should've", "could've", etc. valid short forms of "should have" and "could have"?

Solution 1:

There is nothing wrong with those contractions at all. Your spell-checker, like most, is brain-dead.

Solution 2:

"Should've" and "Could've" are auxiliary verb (http://en.wikipedia.org/wiki/Auxiliary_verb#A_list_of_auxiliaries_in_English) contractions, and are common in spoken American English.

"Things've" is not a contraction of the form described above - "things" is not an auxiliary verb, and is not as common as auxiliary verb contractions.

I will note that I was taught you usually should not use contractions in writing (there are exceptions of course - quoting, an author's style, etc.).

Contractions in spoken English, however, are much more prevalent and accepted.