Can I restore downloaded paid apps after a factory reset? I need to factory reset my phone.
However, I need to know whether I will be able to download
the applications I paid for on Google Play once I do the
reset, and download them onto my phone at no additional cost.

If once I do a factory reset I need to pay again to download
all the apps, then is there a way to back them up prior to
doing a reset, and then reinstall them back again?
Once bought, always yours. New device, factory-reset, does not matter -- as long as your account stays the same. Paid apps are bound to the Google-Account you've purchased them with.

To give some proof: I use several apps I bought once on 3 devices simultaneously. I never was asked to pay again when installing them on the next device. Provided I used the same account, of course.