I would suggest having a look at: How to construct a sparse matrix used for calculating function values at multiple points - #4 by dokken
and subsequent posts.
For your problem, you need to insert the point source contributions in your rhs vector.
For your other questions, it is not easy to help you, as you haven’t provided a minimal reproducible example.
My best guess is that you are using a vector element, and that the point source has not taken the block size of the dofmap into account.