there are so many threads on this forum about training that I suggest you look at some of those to save repetition... Personally, I did the PTC course and thought it was great, but it is time-consuming and will take at least 6 months to complete due to tutors' marking time etc. One word of note - who will you be targeting? If it's not the publishing industry, will you need to know the proofreading symbols that well? I haven't used mine since I did my course (!) as all work has been on-screen or for clients who don't know the symbols so I have just had to do red pen marks and 'plain English comments' in the margins. The extra content in the course was brilliant though and therefore not 'pointless' but it is something for you to consider. It's also not cheap, but it gives you the most points towards membership of the SfEP - have you had a look at their website? A good resource in general and may help with some other queries you're having at this stage of your venture.
Also, although it's a bit of a comic way of looking at it, there's a post from March 1st titled 'should I work for free?' from Gpuss that is worth looking at. Different people will have different opinions but I would generally say don't do it. Value your time, especially if you have a FT job anyway.
With tax/NI, yes, you need to register otherwise you'll get done and fined. If you have any element of web presence, you won't be able to hide from HMRC. Register as soon as you start 'trading' and if you continue to do your full time job and think you'll earn less than about 7k from the self-employment, you can get an exemption form for NI contributions. It will possibly work in your favour when starting out anyway as if you don't earn a lot but can off-set things, you may even get a rebate on the tax you pay through PAYE. My suggestion is to sign up to go on the free HMRC tax/self employment workshops - they were invaluable to me for information, not to mention I got a client at one!
Websites are very personal and what you include/exclude will depend on your specialisms/web skills/personal preferences etc. I did a lot of research of other proofreaders' and copy editors' websites first, then made mine, then sent it to friends/family/editors/people in the industry I know for feedback but whether it's 'good' is subjective and only a client can answer that. Yes I have had work through it, but nowhere near the amount of clicks I have had. But who looks at it?! Think about what you want the site for and work from there...
I hope some of that helps, but I really would recommend you take time to look through some of the older posts on the forum as there are some real nuggets of information and help in there.