Raised beds

1. I wouldn't. Put large angular gravel under the sleepers for drainage and levelling.

2. 20cm is fine, on top of the existing topsoil with turf removed. The object is to create deeper, better drained topsoil that's possibly slightly easier to access (if the beds aren't too wide). Don't put membrane down, just remove the turf. Weed seeds arrive by air, they won't grow up through 20cm of soil.

3. No, but you will need to connect or brace the sleepers to stop the soil pushing the walls out (although this is not necessarily much of an issue with heavy sleepers and heavy soil). But put membrane on the inside of the sleepers to keep excess water off. They will eventually rot, but are easy and cheap to replace.

FWIW I used 3 x untreated oak sleepers for a few raised sections of bed, connected with galvoband and tek screws. They're going nowhere and look a bit nicer than softwood, but are a little more expensive.

I agree with the above re the sods or at the very least remove the weed membrane or save it for the top where you will have weeds

I would go Phormium over Cordyline as the Cordyline is cute when small but once they get stalky they are unattractive I think. Buy one of the smaller Phormiums or it will be too big for that bed in no time, I'd put it in the corner roughly. I'd definitely put in some bulbs for spring colour, couple of heathers maybe and I love Heuchera, can get fab colours in them and all year round colour and don't grow huge. Couple of smallish grasses so you have a bit of movement. The perennial geraniums are good too, come back every year and add a bit of flower colour.

You could put weed fabric on top, cut holes, plant plants, gravel top maybe. I don't like bark, well in fact I do like the look of it but birds will dig in it and throw it out onto the grass all the time which annoys me!

No grass will not grow up thru thick layer of top soil. Yes was also going to recommend perennial geranium ( common one is Johnstons Blue) not to be confused with pelagonium geraniums. On their own they will take over the bed and also cover bare soil and mean less weeds. . They would be good start as smaller plants really 'ground cover' until larger get established. Bulbs a good idea also if sunny spot and will have done their thing by time geraniums get going. Now is time to put in bulbs but time enough for other ones in spring.

Cardboard box would be perfect there instead of the weed membrane. Will kill of the grass and decompose over time allowing whatever you plant root down

That will be a pain to mow! Square off the corner with more wood around the tree and decorative stone. Or edge the lawn in a curve from either side of raised beds around tree with one of the many products available and stone again.

How big is that tree going to grow? If a good bit larger I wouldn't put it in the corner as it will just be overhanging the other gardens and a nuisance if they want it trimmed or whinging about leaves etc.

It is possible move trees, I moved one about 3 times before I was happy with it but it's a risk. If I was moving it at all I'd move it away from there altogether so that your raised beds are more visible however you may have it there for screening purposes but now that you mention it it's definitely going to be a pain from a mowing point of view. If it's one thing I have discovered over the years never plant trees randomly stuck in the grass, always a nuisance for mowing unless you do a bed or enclosure of some sort around them. Hence the moving of my own tree 3 times, it started off in the lawn, bad job! Now it's a beautiful tree incorporated into the side border.

Don't build anything up round the tree, it looks planted a bit deep as it is.