You can stay however many days you can afford, whether it be using points or cash. If you are a DVC member, you can reserve rooms at a discounted rate.
Seeing as you mentioned that you are renting, you have a couple options:
1) Will you be purchasing annual passes? Because depending on the time of year, you may have the opportunity to reserve a room at a discounted rate.
2) You can reserve a room yourself just as you would if you were booking a whole week, except you will only be reserving one night.
3) and probably the best/safest bet: You can have the member whom you are renting points from reserve a "Cash Room" for you. This means that you will get the discounted DVC member room rate AND you can stay in the same room as you will be staying for your point stay.
We did this in Dec. for our own room - we didn't have enough points for our stay, so we paid cash for Fri. and Sat. nights and used points for the days before and after.
The best benefit is that you don't have to check in and check out of your room or into another.
Have fun and congratulations!:wave: