This is not so much an answer as a bit of guidance to start with and some things for you to think about/answer.
The first thing to do is cut down all the overgrown grass and weeds, then you can see what you're dealing with, and it would be useful to see more pictures once you've done that. It does look as though any grass has a high percentage of weed; if its more weed than grass, it may be that you need to take it all up, decide where you want the lawn to be, prepare the area and lay new turf. However, given the difficulty in getting supplies for the garden currently, you may have to wait to do that till later. The angle of the sunlight makes it quite difficult to see the photos clearly too; the dark area in front of that door (what is that?) is impossible to view clearly to see what's there. Then clear away any clutter lying about if you don't want it, or gather it together somewhere out of the way if you do. Whatever that is hanging over the fence from next door needs cutting back too, can't tell what it is from the photo.
In respect of the wood base, take it up and get rid of it, that's the only way to get rid of the mice and rats (when you can, given the lockdown in the UK). Do you know if there's a hard base beneath, like concrete? Taking up the wood will reveal if there is anything else under there. Were you intending to have a shed? It looks to be a pretty small area, so knowing just how wide and long the main area is in terms of measurements would be helpful. Lastly, what's behind the green bin? There's light coming through there and I can't tell if it should have a fence because the other side is next door, or whether it's more of your garden behind it.