Update: I think 3 is possible:

I think 4 is possible. I believe this has a unique solution. (Oh, just saw that someone posted something similar)

Update: I think 3 is possible:

Solving this is as follows:

First, add the diagonals and the 8 is then forced.

Next, there's only one way to connect the 12 and the 16 without creating an island:

Next, there's a deadend, that must have a 1

And you're done

I think 4 is possible. I believe this has a unique solution. (Oh, just saw that someone posted something similar)

I think 4 is possible. I believe this has a unique solution. (Oh, just saw that someone posted something similar)

Update: I think 3 is possible:

I think 4 is possible. I believe this has a unique solution. (Oh, just saw that someone posted something similar)