You ask "can I" well this would depend on your qualifications in electrics and if you are able to sign the paperwork. Also location Scotland it seems does not have Part P. If you mean could a qualified person then yes they could use power from the ring main with a suitable fusing arrangement. As to cable approx 1mmÂ² is normal but with no information as to how the cable is to run or what type of lights you intend to use it's like asking how long is a piece of string? Because your in a kitchen the Part P regs in England and Wales require you to notify the council for most work undertaken which is why even kitchen fitters now employ electrical firms to do all electrical work.
This would depend on the electrician checking your work. If he follows rules to the letter he can't sign the design and installation parts of the form only the inspection and testing parts. I would be prudent to work with an electrician not try to find one later and work as his laborer rather than designing and installing then looking for someone to pass it. There are many rules and regulations as to how far under the surface it needs to be if protection is not provided how close to corners etc. It would be too easy to miss something to which the tester is aware and so not get it passed.
Part P is the law that says you must follow the advice given by IEE/BS book called BS7671 but this book was never designed to be law as a result there is a lot of interpretation involved. To answer your original question most cable buried within walls is called twin and earth and is a flat gray cable which is in turn normally protected by capping or conduit the former normally used before plastering the latter after as it requires less plaster to be removed. If the cables run in places considered as non standard they may need better protection. The guild to 16th edition (also called BS7671) is not too expensive and does explain areas where extra protection is required. I am not sure how the pre-wired kitchen units fall into the new rules I know that units with plugs and sockets which interconnected each unit and one unit would be plugged in were available and I think they get a type approval so getting around the Part P but I am sure someone will post advice on pre-wired units. And if you get an electrician to design and agree as to what work he will allow you to complete for him he will make sure you don't fall foul of any missed regulation.
Firstly, before we get carried away!
Do you know a part p sparky that will sign the work off for you?
If so, i'd contact them first. Don't assume an electrician will do this, because they are not really supposed too. If you wish to do the work yourself, you can apply to local building controls (they will charge for this). Then you will need to prove you can design, install and test and inspect the system you are wanting to install. As for taking a spur from sockets for lighting do-able but not good practise.