Male bettas really prefer to be housed alone. Some will tolerate some types of fish as tankmates, some are too shy/anxious and get stressed with tankmates, and some are too aggressive and pick on tank mates. In your case, your betta may be stressed by the presence of tank mates.
Also, some other fish will nip the betta's fins, and zebra danios are one species that does tend to be fin-nippers. Also, danios prefer cooler water than bettas, and need more room for them to school, all making them "maybe OK but not recommended" as tankmates for a betta. He might bite his own fins/tail if he's stressed. Check to see if he has any scallop-shaped pieces missing from his fins, and if so, he needs to be housed separately.
77F is minimum temp for a betta. He'll do better at 78-80 on a normal basis (not fluctuating between 78 and 80, but a set temp in that range), and often heal faster or deal with illness faster if the temp is 80 - 82.
Waer flow is important, and may be too strong, as indicated above. In the wild bettas live in relatively sluggish or slow moving streams, marshes, etc. The bettas bred for aquaria are long finned, and the long fins are difficult to maneuver if the current is too strong.
I suggest feeding at least daily. I feed my betta and my other tropical fish at least twice a day, sometimes 3x/day. He needs specific betta food that lists fish as the first ingredient, and pellets are better than flakes, as flakes tend to cause bloat. Freeze-dried foods also cause bloat, even though they are frequently marketed as betta foods. If you want to feed blood worms, brine shrimp, etc., it should be live or frozen (thawed), not freeze-dried.
He also needs frequent access to the water surface, as bettas are labyrinth fish, meaning they breath air. He can breathe underwater, but most go to the surface for gulps of air on a regular basis. If he is staying at the bottom it may be due to presence of other fish between him and the surface, as he may be intimidated by them. You mention that your tank is tall - remember that in the wild most bettas live in shallow waters. He may have trouble getting to the surface for air if the tank is fairly deep and the filter current is too strong.
A week isn't too long for him to have settled into a new home. Many bettas don't eat for the first several days in a new environment, and he can survive more than a week with no food. On the other hand, if he is in any way sick or in a weakened condition, he may need to eat sooner.
Personally, I would recommend putting him in a separate tank of at least 2-2.5 gallons, though 3.5-5g is better, and a 10g isn't too big. An inexpensive alternative would be the largest size of these
Petco Pet Keeper for Aquarium Fish - Temporary Fish Tank and Betta Fish Tank from petco.com. The large holds about 3.5g of water when totally full, around 3g when you account for fish, filter, heater, substrate (if you want any), etc. If you don't have another tank and don't have the resources to get one, anothr option would be to build him a separate area within the larger tank. This could be done by using a plastic mesh basket such as those used for pond plants, attached to the side of the aqarium so the top is above the water level, or by building your own container out of plastic mesh (buy at craft stores like Michaels or Jo-Anns). It comes in sheets which can be cut to size, fastened to each other with fishing line to make a box, and again attached to the side of the aqaurium so that the top is out of the water. The enclosure needs to hold at least 1.5g of water (about the size of a shoebox), and again, bigger is recommended. He will need some hiding places in the enclosure with him, like plants, caves, etc. Java moss would be a good choice, as it is very lightweight and he can rest in or on it. Floating plants would also be appreciated, as bettas don't like very bright light and by bringing him to the surface of the tank he will be closer to any light source above the tank. You may want to attach some moss or other plants (live or silk) to the sides and bottom of the enclosure so he isn't stressed by tankmates coming to check him out through the mesh. If you place your heater in or near his enclosure he might get a bit higher heat than the other fish in the tank, which would be good.
Good luck with him, as well as with the rest of your fish.