Yes, working against any significant form of resistance can be considered strength training. You can get in a great workout using just your own bodyweight for resistance. Take a look through the Spark exercises for ideas www.sparkpeople.com/resource/exercise_demo
You should rest your muscles at least 48 hours between strength training sessions - realistically this implies 2-3 strength training sessions.
You should be aiming at an all-body workout that works most of the major muscles. There are 3 major muscle groups (upper body, lower body and core) and muscles generally come in opposing pairs. By choosing compound exercises that work several different muscles at once (eg. squats, pushups),you can probably get in a good all-body workout in 5-6 different exercises in 20-30 minutes.
Basically, strength training is all about quality, not quantity, and it is far more effective (and not to say faster) to genuine challenge your muscles in just 4-8 reps, than to do 20-30 reps with easy exercises or light weights.