Was the road just south of the Gothic Chapel the one that was closed and weed covered?
South being a relative term, yes.

Here's a picture of what this gate looked like in July:

And below this text is a photo of a sign that was near this gate. I can't read it, but maybe someone here who reads cyrillic can translate it? It *might* reveal why this gate is closed or it might be as simple as a message that reads, "stupid tourists, go home!"

Do you know how the rescue station is reached?
I don't know! I've never heard of this location.
Is the entrance to visit the Cottage through #57?Can you not follow the road path from the Cottage?
Yes, the main entrance to the Alexandria Park is #57 and if you enter through this gate you should be able to follow the pathways to the LP. My problem was we were in the Peterhof complex... There's also one other gate, but I can't verify that this gate is open. It's located just "North-West" from the Gothic Cathedral.
Also I am lost on #61 House of Konstantin. It looks small to be for a member of the family so I am wondering if it is named after a faithful courtier or servant.
I don't know the answer to this question either! Sorry!