Yes the rent price will be fixed for the length of the lease. You can always try to negotiate the price down a bit although demand is usually high enough that landlords won't budge.
You may be offered a six-month or one-year lease. If you really like the flat, and think you have a good deal on the rent, you could ask for a two-year lease with a one-year break clause (i.e., after one year, either you or the landlord could cancel the lease).
I personally always try to rent direct through a landlord and avoid estate agents, they charge outrageous fees and usually offer very poor service.
Don't ever go with an agency that makes you pay to view flats or see property lists.
Make sure your landlord puts your deposit in a deposit guarantee scheme, they are required to by law. It's great, it means that when you move out any disputes about the deposit are handled through a neutral third party.
Many landlords are not keen about pets, but you can always offer to pay an extra pet deposit for any extra damage, and say you will steam clean the carpets and such.
Make sure you do an inventory and flat check when you move in, and make note of any little dents or flaws, so you don't get charged later.
You generally pay all the bills but some may be included (like water).
Read the lease very carefully! Make sure it specifies that you have the right to quiet enjoyment of the flat and the LL needs to give notice to come by for non-emergency purposes, and that the conditions for breaking and renewing the lease are clearly spelled out.
Good luck!!