xperia x10i problems

[mo_d]

Hi, i have an xperia x10i. It often freezes and restarts, often I get a burning smell from the back of the phone. It restarts so much, often at least believe or not, at least 50 times a day. On Saturday, I actually lost count after 54 restarts. On other times, it does not start at all. Most of the restarts seem to happen, not just  when I'm using big apps, which is very rare now but problem still persists. But it also happens when I am using the browser and a lot of times, just when its locked or just on the home screen. It could happen and freeze while the phone is switching on, when its charging, when I'm actually texting or even when its switching off. The problem is I don't know if there are any service centres near me, the locator of service centre does not seem to work, so I have a few questions:
1) Do you have any idea why my phone freezes or restarts and has a burning smell?
2) I don't have a proof of purchase, I bought my phone online( I can provide proof of that only), so will my phone be able to be repaired?
3) Is it possible for  to arrange for a courier to arrange a pick up from my house?
4) Can you find out if any service centres are near me? Please find out for me. I live in Lancashire, UK.

The phone is factory unlocked and unbranded. PLEASE HELP.