Ahh the perennial issues. Although I do a bit of locations work the answer to some of these questions are still somewhat elusive..
The general answer is.. it all depends. Technically, yes, you do need permission from the council to film in a park. But by that rationale, someone with a camcorder filming his mates playing football would also need permission, which doesn't make sense. If you're relatively small scale, I'd just do it, don't worry about it. If you're stopped by anyone just apologise, say you thought filming in public spaces was allowed, and tell them it's for an exam piece. Also explain that you've taken steps to make sure you filmed when it was fairly quiet so as not to get in anyone's way, and ask their advice as to where might be a better corner of the park in which to film. You just have to flatter people sometimes, and act a bit ignorant. I struggle to blag it when there's ten trucks full of gear and 50 crew trampling the place with extreme disrespect, but I have still blagged it even in those circumstances.
Running past shops, yeah no problem. No need to ask for their permission. There's specific rules on background buildings, no permission needed unless there's a direct reference to them. Eg, wide shot of actor walking downthe street, he's just been to a crack den. That's fine. Compare it to this alternative scenario... - shot of actor walking down the path of number 19 in the street, he's just been in the crack den. See the difference? Obvious really.
Train station - hmm, that's a much trickier one. They are generally quite tough to film on. If just using dv cam then maybe try to blag it as much as you can. If you're there in numbers then you might have a bit of difficulty. Pick a quiet branch line if the story allows, I've done some stuff on a North Norfolk line and that was fine, didn't need to go any higher than a regional manager. I've also done Charing Cross for an hour and going with the flow of commuters etc, and that also worked out fine, but it did cost £500. Ouch. I think The Bourne Ultimatum at Waterloo was costing £5000 an hour or something, but I've also done a bit at Waterloo for substantially less than that too. But generally speaking, tricky. I'm just about to make use of another private line for a film, and that will hopefully work out ok, but will also cost a bit.
Weston beach.. I would just blag it - beaches have various ownerships but most are controlled by the local district council. Again I'm just about to oranise filming on many beaches in Lincolnshire and have pre-secured permission from the council at no cost (although we'll actually be making a donation to the Lord Mayor's charity). The bit of beach that is exposed when the tide goes out, ie the land between high water and low water is actually owned by... the Queen!
There is a "form" of sorts - it's a standard filming location agreement - for really straightforward stuff I have a much paired down version I could send you.
Finally.. insurance. This is most definitely a grey area. Again it very much depends what you're doing and what you hope to do with the finished piece. In theory you ought to have public liability insurance but this is stupidly expensive. I had to pay £700 to insure the "making of.." documentary for "Dead Man's Shoes" even though we basically interviewed Shane with a dv cam sitting in a room, and then followed him round at Edinburgh. Groovy.
On balance I would say just shoot what you can without worrying about it. Just use common sense, maybe set out a bit of a written plan for what you would like to do. I have to do full risk assessments for every location we film at, which is tedious in the extreme - I can also send you a paired down bog standard risk assessment. Even if you're not insured the fact that you have taken risk into consideration and acted to reduce it will be favourable. Councils in particular (if you do have to speak to them) will look more favourably on you if you have done a risk assessment.
Ok another "finally" - the stuff you raise about insurance etc for this sort of no budget filming is a good point. If I were you I'd think about contacting your local regional film office and asking them what they think about it. Yours is Southwest Screen -
www.swscreen.co.uk tel 0117 952 9977. I don't have the contact name to hand but I can get it for you. If you phone them and tell them you're doing some no budget filming in the region which is getting a bit more serious, thinking of doing a bit more, but were just after a bit of advice to find out if there's anything you ought to be doing, then that puts the ball firmly in their court.
I genuinely don't know the answer here as all of the stuff I work on is covered by insurance, so I'd be interested to know what the film office thinks. In fact I might raise it with a few people I know elsewhere. SW Screen's role is to foster film making talent in their region and it would be very poop indeed to think that the talent is pole-axed at the first hurdle by being unable to film anywhere due to lack of insurance. Hope this all makes sense.
Summing up - just keep shooting stuff - but using a little forethought.