New Betta Fish!!!

For a small ammount of water light his i'd get a 25W heater (like the visitherm stealth). If the one you have was used in a 20G it might be a 100W and incase of a failure you'd end up bringing the water to +100 degrees or more... not a nice thing to do. The VT stealth can be found for 12$.

I think the filter you described is an undergravel. Could work for this setup - any filter is better than no filter at all. Just make sure there is a spot without current where he can rest - or a cave or something he can rest in.

I have mine in a 10G with a regular HOB filter. My betta intenionally swims into and plays in the current even though many people claim they dont like current or waterflow. I think it comes down to the betta having time to adjust to it - after all he spent most of his life in a small cup - so once he gets in a bigger tank he has to develope some strength to swim in a current.

Besides that - i assume you know about the nitrogen cycle and waterchanges because of your goldfish - but in a small tank like his and while its new (and uncycled) i'd do 50% waterchange every or every other day depending on test results - you really dont want ammonia or nitrite to spike and cause him harm.
 
When I got him at the pet store he was in a coffee cuo sized bowl. i felt soooo bad for them. I wanted to buy them all. At first when I got home and put him in the tank I had no light or filter in there (my lightbulb blew). He wasnt moving around very much and I thought he was already sick, but when I did the first water change that is when I added the filter and the light. It was like he woke up and was in a whole new world. He started swimming around and goin in and out of the bubbles. So I thought he liked it. I will porbably stop by the fish store on the way home to see what I can find.
 
it's not a bad idea to put some "furniture" in there for him but be careful with an under gravel filter(UGF) the gravel bed is your filter so the lage objects blocking the flow of water can cause anerobic pockets underneath- so everytime you do a water change shift your "furniture" around.
 
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