1 Most US employee share schemes are non-qualifying as far as HMRC in the UK is concerned. That means that the difference between what you pay for the shares, if anything, and the price when they vest is chargeable to income tax and national insurance contributions. There is no way around that, that's simply the way they are taxed. Most people don't have the cash to pay the tax when the share vest so its normal for the tax and nic to be covered by selling some of the shares. If you held on to them and paid the tax and nic from your own resources, their cost for CGT purposes would be the sum of what you paid for them and the amount on which you paid income tax and nic. If at some stage in the future you sold them for more than the CGT cost, then you may have CGT to pay.
If you sell shares at below the amount you paid for them, ie the vesting price was lower than what you paid for them, then youn will have no income tax or nic liability.
2 That's correct.
3 That's correct.
4 If the transaction is run through the payroll, then the income (the gain on the shares) will be included in your tax year end P60 so it is simply reported as employment income. Any capital gain, as mentioned by you in 2 above, would be reported in the capital gains pages of the tax return.
If shares vest after you leave the employer, the gain will still be suibjected to income tax. I tend to use boxes 3 and 6 in the share schemes section on page Ai2 of the SA101 pages here for clients.
I hope this helps but let me know if you have any further questions.