Im having trouble with my black moor and the tank :/

Question
Hey, my name is emma. Im thirteen and just recently (two days ago) got a 10 gallon fish tank for my room, along with a fantail goldfish, a black moor goldfish and two golden snails. i have a few questions and problems so here they are.

1: (Black moor) My black moor isn't eating anything i feed it, I've tried blood worms and regular goldfish food but he spits it out. He spends most of his time at the top of the water breathing in air and not water so do i need a filter? And what kind of food should i feed him?

2: (fantail goldfish) Same question and problems as seen above.

3: (snails) my snails always make this white discharge(white slime) and i think it may be affecting the water, because this morning i woke up and the water was suuuper foggy. Is it because of the snails?

4: (tank) do i need a filter and an air pump?(with a bubble stone)

 I hope you can help :)   Thank you

Sincerely,
Emma M.

Answer
Emma,
The smallest size tank for 1 goldfish is 20 gallons with 10 gallons for each added goldfish. So, to house your fish correctly, you will need a 30 gallon tank, and here is why. Goldfish do not have stomachs, and food comes right back out as waste. The waste is ammonia first. If not taken care of it turns into nitrites. Both of these can kill your fish. If your fish is at the top of the tank, it is telling you there is not enough oxygen in the water. You can change that by adding both a filter and an air stone. Because you have them in such a small tank with no filter, you will have to do water changes daily. You will do this by vacuuming the gravel and removing 30% of the water replacing it with clean conditioned water. You have to use a water conditioner that removes the chlorine from your tap water. The cloudy water is because there is no filter and too many fish. When you add the filter, your tank will start the cycle process. When this happens, you will have to test your water with a liquid testing kit API makes a good one. You will have to test your water daily for ammonia and nitrites and then do a water change depending on the reading. This will have to be done every day for at least 6 weeks while the filter cycles. If you do not test or do water changes, you run a very high risk of losing your fish.