This answer varies wildly depending on the size and type of aircraft involved, but in general:
How is the runway cleared?
They remove the aircraft, then sweep the runway for smaller debris and then inspect the tarmac to make sure the aircraft didn't gouge into it, and tear up chunks.
Is the plane towed away (scraping along the ground)?
No. The aircraft will usually be lifted and placed onto an oversized platform with either cranes or hydraulic Jacks, and then towed. Dragging it along the tarmac would damage the tarmac (more so then the landing already did) and do further damage to the aircraft.
Are there cranes available in large airports?
Gear up landings (and catastrophic hard landings where the gear collapses) are so rare that no airport (that I'm aware of) has the heavy duty mobile cranes required standing by. It would cost way too much to maintain them and keep operators for them on the payroll when they'd just sit and rust for years and MAYBE be useful for 3 or 4 days. It is much cheaper to hire a crane and operator.
Can a single crane lift and move a large airliner?
Maybe; aircraft manufacturers will often publish manuals for how to recover and salvage the aircraft, and the crew removing the aircraft would refer to those.
Is the plane considered a write-off or can some be repaired and continue in service?
Not necessarily. This part varies depending on the age of the aircraft, how hard the landing was, and a million other factors, but the aircraft can usually be salvaged. The repairs could be astronomically expensive, and the work could take several months if not a year or more.
Where would such repairs take place?
This also depends. If the airline has a hanger at the airport where the plane is being salvaged from, the answer is simple. If not, that's when it gets tricky and expensive. Usually this means renting hangar space (which is SO much more expensive then any other commercial space) and either flying in your own mechanics or hiring a local crew. The necessary repairs and inspections would be done to get it to the point where it could safely fly back to an airport where the airline has the personnel and equipment to finish repairs.