I included the things I really didn't know.

1) May I borrow your high heeled shoes? (Give permission) Yes, you can (may?)
2) keep far (move away, stay away are possible?) from rocks and trees.
3) Hamlet doubts whether or not to kill Claudius.
4) The theme of false appearances is linked/related to/connected, associated with that of equivocation. Are they all possible?
5) take a teaspoonful of cough syrup before bedtime.
6)I had a satellite dish installed to watch English channels. I can get (receive?) the BBC now.
7) I use a remote control to turn the TV on and off.
8) My son sometimes snatches the remote control off my hands and switches over to another channel.

1. OK -- and use "may" not "can" -- "may" is for permission; "can" is for ability.

2. All are OK.

3. OK

4. All are OK, yes.

5. Capital T at the beginning; the rest is OK.

6. OK

7. OK

8. Use "out of" rather than "off"