Yes, it's possible! Unlike larger livestock (sheep, pigs, goats etc.) you don't need a CPH (Country Parish Holding) number. If you have 50+ (won't apply to you!) then you also need to register, but that's not permission, that's just letting DEFRA know. The only potential legal problem that you could run into is if your house deeds forbid it.
I have close neighbours on three sides, and none have never complained. This is despite the fact that one has spent years complaining about our leylandii, although another neighbour went off us at the same time (slightly Hyacinth Bucket characters that lot, not a massive fan of them to begin with tbh), although they never actually said anything. The final side (despite a change of owners, and incidentally the house whose garden is bordered by the coop) has always enjoyed their presence.
Don't keep cockerels, as they will annoy, but hens are really very quiet. To avoid rats / mice, don't scatter food on the ground, keep the feeder in the coop overnight, and store food in a metal dustbin / something else they can't chew through (not plastic!). Urban foxes are the biggest potential blight. Make sure you shut them in as soon as it gets to dusk (if you won't always be in at that point, it is possible to get self-closing popholes - they put themselves to bed when it gets dark, although you should check to make sure; if they don't go in by themselves then it's a sign of red mite), don't let them out of the run unsupervised. I never experienced an attempted dig-in, although I don't think I've ever seen a fox where we live (large village / small town), compared to the millions of the things in the city. What I did experience, however, was the grass getting a bit too long by the run, the wood deteriorating slightly (the coop was about 6 years old at this point) and the foxes found the weak point and got in like that. Bastards took the lot. Really, it's enough to turn this veggie in favour of fox hunting! If all else fails, there's always the option to buy a humane trap and a .22 gun (no licence required, according to the advert) from the back of Country Smallholding magazine . . . yes, I do know someone who's doing this in the middle of a small town! One option is to put a chicken wire "skirt" around the run which the foxes can't dig through, or to dig chicken wire in (either several feet deep as a 'wall', or under the entire thing).