Hi Brendandrage
1. I had a tap on a previous kettle but I now use a jiggler syphon and silicon hose with my new kettle to transfer the wort (hot and cold). The tap was easier and quicker to use but I took it apart and cleaned it after every brew and this took time. The jiggler and hose works perfectly and it's easy to clean. The negatives are that you have to wear protective gloves when transferring hot wort and it's sometimes a bit clumsy to use. If you do a whirlpool after a boil you may disturb the trub when you start the jiggler. Here is an alternative
http://www.biabrewer.info/viewtopic.php?f=53&t=1108 method of draining your kettle. You can also use a racking cane (copper pipe is fine) or, if you only plan to transfer cooled wort, a plastic auto syphon.
2. False bottom. I use a hop sock so I don't need a false bottom in order to prevent hops from blocking the flow of wort when doing the transfer. I do however have a cake cooling rack in the bottom of the kettle to prevent the bag from melting during the mashout procedure. A false bottom will/might(?) serve the same purpose.
3. I had a pipe from the tap to the bottom of the keg because I would have "lost" about 2l of wort on a 20L brew if I did not extracted every last bit from the kettle. I ended up with all the trub in the cube but this was not a problem as I could leave the trub in the cube when I transferred it to the fermenter.
You will get a lot of different answers/opinions on your questions and it will be pretty much up to you to decide on the best options for your situation.
Good luck, and let us know what you've decided on in the end!
L