Hi
I'd say about four inches of good soil would do it. Grass does not really need a lot to grow but the better the soil the better its ability to hold nutrients and water. With that you get a more dense root sytem and healthy plants that can outcompete weeds.
You have to make sure that the soil beneath the top layer is free draining. Particular after new build you can find a lot of compacted soil or builders rubble buried. If the grass is in a constant puddle of water it is likely to not perform well.
4m x 2m is not a particular big area to cover so from a management point of view more than doable.
However, some points to think about.
As mentioned before, water.
At the very least you need to water until the grass is established. Once that is the case you can probably restrict the use of water, in California the whole country side turns brown during the summer and springs back to life when the rain comes in autumn. But I imagine that you envisage a green lawn all year round so you need some kind of watering system. Since you are talking about such a small area I'd say a hose with some kind of sprinkler head would probably do it. Probably depends on the water pressure. Or maybe a more refined system with some sprinkler hoses running along the edge of the grass and sprinkler heads aiming inwards? Simple enough and won't break the bank. But there is the cost of water!
It is important to have a grass variety that is suitable for your climate. If you get seeds from a supermarket or garden centre keep that in mind.
Any particular reason you are thinking about using plugs? As far as I understand it they are used as a cost cutting measure instead of roll out sods ( which is what I would be going for) and seem to be a lot of work. I could imagine that you have to wait a while before you see results, as well.
Grass suitable for drier conditions have usually firmer blades which can be uncomfortable to walk on bare footed or lie on, just to keep in mind.
Having written all that, I personally would go for a hard landscaping option such as attractive tiles or gravel, maybe broken up with some larger pots on the edges and planting holes in them.
Regarding the raise bed, the use of railway sleeper, once so popular, is pretty much frowned upon nowadays, particular when used close to edible products. Their carcinogenic nature makes them unsuitable. If you want to go with wood I would go for similar sized untreated beams, the solidity of a beam adds a nice structure to a garden, a bit of gravitas. Keep in mind that wood is an organic material and sooner or later needs replacing.
1 meter height for the raised bed is quite good, it eases access, not so much bending over. Maybe a bit of variety in height? Lets say two or three levels, one for herbs(high), one for tomatoes and such(low) and something inbetween for flowers, that could look spectacular. ( I would mix useful with decorative anyway)
You need about 30 to 40 cm of soil in the raise bed, depending on what you want to grow. Herbs in general are not spoiled and do not require much at all, some thrive under harsh conditions. Use some kind of filling material for the base of the bed but keep in mind water retention and drainage. For herbs you may be able to fill the entire bed with some kind of filling material and keep the herbs in clay pots, a bit like a gravel garden. That could look very well.
When I make up growing mediums for beds I use a third sand, a third compost and a third well rotted manure. For plants like Rosemary, Lavender and such I would use a lot of fine gravel around the roots to ensure drainage.
There is quite a lot of material to shift. The raised bed alone is around 5 cubicmeters ( depending from where you take the measurement for the 1 m widht)That's a lot of wheelbarrow loads.
I don't know what your area looks like, of course, and how you can access it. I would put in the raise bed, prepare the ground for the grass and put whatever stones that come out as filler in the raised bed. Fill up the raised bed with additional filling material and soil, make sure not to compact the ground for the grass to much, maybe lay out planks for the wheelbarrow. Then do the grass.
Hope that helps.