A bottle trap or pear shaped trap is not a good choice for a kitchen sink, because of the types of things that go down there. Even if one is to be used it should look more like the first picture, ideally the second installation is better. No added bit at the beginning on either installation.
The receiver for the stat is now recommended to be fitted externally as the boiler front cover is metal and also cause a problem. Its not a difficult task to move. Do you still have the blanking plate for the boiler? your gas engineer should be able to mount this for you.
1 bar is not enough it should be 3 bar for domestic hot water. There should be no water dripping out of the pipe. After it is fully repressurised and you still have water dripping out, it will be another problem and you will need a qualified G3 engineer.
If your house is a 1950s house the incoming main could be lead. If that is the case its unlikely you will be able to change the tap yourself. It might be worth getting a plumber in to look.
Our waste fittings are quite snug and if you are only using it for your laptop stand they would be fine. I looked at your link they would do the job and would not need to be glued.
It will then depend how confident you are with your Diy, its something you can do yourself. You might need a few extra bits if the bolts underneath are rusty. Make sure your isolation valve works. Everything is clean and go for it. I don't know the valve that you have mentioned. You might have...