Since your just starting, there might not be too much of a change bacteria wise if you change your gravel although I will still watch my levels. (I forgot, are you cycling with or without fish?)
If you dont have fish, I will drain most of the water down and take the gravel out, and then wash the sand until the water runs clear, put the sand it, fill it back up with dechlorinated water. Make sure that while doing all this, your filter media are staying wet.
If you do have fish, remove all fish, put the fishes in a bucket with a heater and maybe a small bubbler...then follow the steps above.
When I changed over to sand, I didnt let the sand settle and it did break my impeller/motor on my AC70 (AquaClear) and had to change that part. What I did to prevent this from happening again is I put a prefilter on the intake. I just got some sponge media and cut it to wrap around the intake with a rubber band. This will stop the fine sand particles from going into the impeller/motor. Also make sure that your filter intake is not too close to the sand that it will suck it up. I have mines about halfway down.
With filters, I like to double the recommendation. Lets say, I have a 29 gallon and I use a 70 gal filter. So for a 20 gallon, I would try to find a 40 gallon filter or more. Try to get one where you can control the flow like the ACs so your fishes won't be blown away.
In regards to bigger filter, faster cycle, I don't know. Maybe it will help, later on because you will have a bigger space to put filter media on which means, there is more space for the bacteria to grow on. But on faster cycling I dont really see how a bigger filter can help.