how much will it cost for a new front door?

Depends upon the type of door. You could spend as little as £50 or well over £1.000. Installation could run anywhere from £100 to £500 or more.

Not enough information to give a better estimate.

The best replacements are called 'composite' doors. These can be had in a variety of styles and colours. They are insulated, secure and need no maintenance.. Not cheap, but will last a lifetime.

See link for examples.Research for best deals.

Just go to the local hardware store and price them out. If you don't have any huge hardware stores nearby then price it out online. Installing one of these yourself isn't hard if you are the least bit handy, but make absolutely sure you get the right size and get one with all the holes cut in the right places and get one that is prehung (already has the hinges on it).

I think mine was about £700 for a UPVC one (had a couple of windows fitted at the same time) It came with numbers on. I used a local firm who fitted it.

Once you get above number 50 they get a lot more expensive

Shouldn't cost you anything. It's the landlord's problem unless they can prove you damaged it.