You could find a 25-30 hp Ford 1710 with loader, or an equivelient Kubota with loader, for $4000 - 5000. These are the compact tractors. They will do everything you want. They will be easy enough for you to operate safely.
The trouble is finding a good one for that money. Do you have a mechanic type friend to go look it over with you? Unfortunately there are a lot of bad folk out there doing horse sales..... A hint is the front axle on most of these older compact tractors no longer have parts available for them, or will cost over $5000 to repair, so you want to _know_ it has a good front axle on it! And so forth.
A rear mounted mower on the 3pt is much easier to deal with on a tractor. They make rougher brush hog type that mow through small trees, or smoother finish mowers that cut like a lawn mower. I don't know which you are looking for. But I would look into those.
An ag tractor is not really designed for major dirt work. You could hire a construction fella to do it quickly in a couple hours and be done with it, or putz with it yourself all summer and still not have it right. Doesn't matter what size tractor you get. Without knowing what you have to do, it is hard to judge. The tracotr will work well for minor things, but if you have a major huge project, it becomes time vs money.....
The compact loader tracotr will be excellent for the 8 foot by 8 foot door. I have that exact size on my barn, and I bought a 1720 Ford/New Holland tractor with that in mind. I can't use a bigger tractor. I have an IHC H tracotr as was mentioned here, but had to duck my head to get in the door, and no live hydraulics, no power steering, and pretty worn out, it was _not_ smaller female friendly - big tough he-man me struggled with it.
So, yes, one tractor will do what you want easily. Used Kubota or Ford compacts are available & can be dependable if they were not abused. The older 1950/60s tractors mentioned also will do the job, perhaps will be a bit more mechanical work keeping them going, but then more parts are available for them.
The issues are: You might want to consider a different type of mower? Placing it on the rear makes things easier.
Your price point is quite low to get a good machine. It's possible, but $7000 would get you a good machine & implements a lot better quality.
Moving construction-quantities of dirt around is not what a farm tractor does well, so it depends just how much stuff you need to do with that job. Might pay to get it proffessionally roughed in, and you do the finishing details if it is a major job.
Here you would need many permits to fill in or modify a wetland area, probably couldn't do any of the dirt work 'here'.
--->Paul