1. At the outset, as there is no legally enforceable right to an easement which would allow automatic access from one garden to another, you can close up this gate at any time if you want. Accordingly, the answer to your first question is that you can place a new fence in your ownn garden if you place any posts on your side of the boundary. In that way, the fence would then be yours and would no longer be on the boundary. The answer to your second question is that you can remove the gate latch which is on your side and secure it closed if you wish.
2. Essentially, the old agreement for convenience is not legally enforceable as it has not created mutually enforceable rights of access from one garden to another. Each party is free to chose whether they wish to continue this arrangement with the gate if they wish. So, if you no longer want it, you can place a fence on your side or lock up the access.
3. Please Rate the answer as unless you Rate the answer your Expert will receive no payment for answering your question.
4. Yes, You can put up the fence, if you want, if you want privacy.