What do these sentences mean? Is the 'coming' and 'using' a reference to future or currently happening?
Do you mind Sam's coming with us?
Do you mind Belinda's using your computer?
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As for the tense, or whether the action is present or future, present verb forms are always used after 'mind,' regardless of when the action takes place. The context here doesn't make the time of the action clear, partially because of the construction and partially because of the use of 'mind.'
'Mind' has numerous meanings and uses. It can be a warning or reminder to pay attention to or be careful of something: "Mind the gap," meaning watch out for the space between the platform and the train car; "Mind your step," meaning watch where you're walking so you don't trip, etc.
Similarly, in the sense of paying attention, it can be used to describe caring for (Can you mind my children today?) or guarding something (Will you mind my bag while I use the restroom?).
We can also use it to ask permission (Do you mind if I sit next to you?), politely ask for a favor (Would you mind passing the salt?), and either display annoyance or let someone know something is bothering you (Would you mind not playing your music so loud? I'm trying to sleep!) or enquire about annoyance or if you are doing something that bothers them (Do you mind my smoking here?).
In the examples above, the intent could be either to ask permission (Can Sam come with us? Can Belinda use your computer?) or to enquire about annoyance or worry (Does it bother you that Sam is coming with us? Does it bother you that Belinda is using your computer?)
The latter seems more likely, and it would be easy to tell which sense is correct, and also when the action will take place, if we were speaking to someone. The situation would then be clear: Sam is with us, or Belinda is using your computer. The confusion here is that we have a Verb (mind) + Possessive Adjective (Sam's, Belinda's) + Gerund (coming, using) form. This makes Sam or Belinda the agent of the action, the person doing the action represented by the gerund; it's not me doing the coming or using, it's Sam or Belinda.
Grammatically there is nothing wrong with it; any Verb + Gerund can have a subject inserted in the form of a possessive adjective, but some people may find it an uncomfortable construction, especially when the verb is 'mind.' There are a couple of ways to say the same thing with less confusion:
If the tense needs to be clarified you could add, "...comes with us today?" or ,"...comes with us tomorrow?"
Here are some links to help: