I would use check in this instance. So the first is correct. Tick the box if you would like more details.
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A rain check (this dictionary entry also has pronunciation) is a ticket that can be used later if a. It can of course also have the sense of monitoring them in a work place or elsewhere. I think check this source code out of the repository sounds better than check out this source code of the repository or check out this source code out of the repository.
In the sentence, tick the box means mark the specific checkbox.
To check out could mean either. Taking a rain check is a polite way of saying no to an invitation, especially in american english. According to oxford dictionary check somebody/something off (north american english) (british english tick somebody/something off) to put a mark ( ) beside a name or an. So the third is not correct.
I think check up on is the best as this can carry the sense of finding out about their welfare. If we have the following. Check out generally means leaving a hotel after settling the bills and also to die. To check in something is to register it.
Check it out usually implies that.
I came across the following example: To check something is to examine it. Check it out and check it can have roughly the same meaning, but check it usually implies a quick, routine check, rather than a thorough examination.