...more fully
1. You need to know enough to tell HMRC if there is a liability or if they have taken too much tax, so although you may not have to file, in order to comply you have to compute your tax liability (although you never report your conclusions). HMRC have been told to be soft on people who ask for explanations, but hard on people who don't declare.
2. If there is a liability or reclaim you request a Tax Return to file, which can be a short form.
>What about PAYE?
3. This is just a payment on account. HMRC don't say it is accurate, although if there is just one employment in the year PAYE will be good enough.
4. In the UK, people seem to value a simple life more than 300-500 reclaim in tax, and so generally don't make claims against PAYE income that they could. They also tend to ignore gains tax, accrued income scheme, pension relief, don't correct the tax position on a change of job, imagine that basic rate deduction on bank interest covers it, and hope that the employer deals with any complication from moving expenses, share options, FURBS etc. Broadly, they ignore the requirement to self assess and rely upon HMRC to tell them they must fill in a form, but never check HMRC's computations (at last internal survey that I have, 17% of these were wrong). However, it works in a painful kind of way.
The tax years in which you arrive in the UK, or change job, or leave the UK are those most likely to throw up a tax reclaim / liability, assuming your annual UK source income is below 40k. Above 50k, most years will need to be filed.
So at the very least, gather your p60 at the year end, plus p11d, plus moving expenses paid for you, plus interest earned in the UK and use one of the online calculators to check whether you have a tax reclaim. You'll need these figures any for your 10-40.
[Some firms including ours will do the above for no charge, and let you know if it is worth your while for the firm to file a UK return.]