Lawns - seeding or turfing

Hello, I am a new gardener. Plan to work on the garden this spring. As you can see the garden has two parts separated by a stone low retaining wall, the smaller part has some grass growing on it but the quality is not very good, maybe just in winter time? the big part has nearly nothing because there were a few big trees just recently been removed. Could you please suggest:

1, I plan to hire a tiller to work on the left big part, I assume I don't need any weed-killer chemicals? In summer there are still some plants on but now there are not many because in winter time and recently removed trees and stumps, it looks bare.

2, how about the quality of the small part (right, around 60m2), grass growing on it but the quality is not good, is it a good idea to remove the turf or use chemicals and a tiller (which I think is easier)?

3, I plan to use grass seeds, much cheaper, this could be easier more than 200m2. Is this a good solution for the big and small parts of the lawns?

4, I live in Scotland, and the minimum temperature at night is still minus -2. is it good to tiller now and then wait until warmer to plant the grass seed?

5, the land is sloped i may also move some soil from the bottom/left to top/right to make it look slightly better /less slope. also may buy some soil, the quality of the soil may be not very good

6, also want to plant a landscape tree (not a very tall one), any suggestions on where should I do it and species? (the left is south)

7, is seeding equally good as turf finally?

Thanks

 

Also anyone can suggest whether renting a bigger petrol tiller or buying a smaller electric one is better? the similar price in total (100-120). I don't know when I will need the tiller again if the grass is planted

 

Your photo is not very clear so if my reply seems wrong, let me know.
I would avoid tilling by machine totally. Its only a small area, which you tell me had large trees and therefore roots. This will frustrate rotovation. In addition, good lawn preparation requires the soil to remain structured not minced. It also will bring up the weed seed bank of ages.
So, what to do ?
I would firstly dig a hole in a typical area, just 1 spade width and depth. Try to remove this sod as a single piece. It will show you what lies beneath, indicating the depth of topsoil, debris, roots, soil colour and give a good idea on how well it will grow grass.
If it contains lots of stones, roots, debris such as glass, metal etc and/or its poorly structured then I would dig it over, removing these. Then rake, level and allow to settle.
For seed you will need to create a fine tilth, for the finished surface will become your lawn. If you choose to turf it, then the extra cost will hopefully buy you the fine surface layer you would otherwise create, but with grass plants established in it.
Before you decide, consider the likely openness of the site, will it be greatly lit by sun or shaded by buildings, trees etc This will guide you to types of grass variety you choose, as will the purpose of the grassed area. Fine lawn ? green carpet? play area for kids and pets. Both turf and seed are available in differing types according to purpose. Hope this helps. Feel free to ask more questions.