Downtown Lisbon is set on 7 hills and, owing to traffic calming measures over recent years, is not conducive to driving.
I strongly recommend that if you need to hire a car only do so when you are leaving Lisbon and pick it up at the Airport or Gare Oriente. Parking is difficult to come by in the old centre and if you have to put a car in a parking garage, you may still have to walk a good distance to reach your accommodation.
Lisbon has an excellent public transport service with metro, trams and buses.
Taxis are not that expensive either.
For apartments have a look at this recent thread:
http://bit.ly/bRLWmI
For Coimbra, I would recommend the IBIS Coimbra.
Stayed there earlier this year. A few blocks for the Centre and the River is just over the road.
Part B:
Cascais and Sintra are easily accessible from central Lisbon via Cais de Sodre and Restauradores Train Stations. Travel time less than an hour each way. You do not really need a car. There are always taxis at the stations.
Evora and Elvas will need a car as there are few public transport links between the two.
Why not leave the latter till you leave Lisbon. Drive from Lisbon to Evora and Elvas? Go to Elvas first, then Evora. Stay at the IBIS In Evora and then drive to Coimbra, (Provided you are prepared to pay the tolls, ) via the A6, A13, A1 motorway thus bypassing Lisbon.
