Tank Water Advice

You'll find most people recommend water changes of at least 25% every week, unless you have a very heavily planted tank like Richard W's.

For 6 neon tetras I would be feeding 3 or 4 flakes a day, crushed up so that there's at least one bit for everyone. If your food is pellets rather than flakes, have a look at the size of a neon's eye. The amount of pellets should be the same as six eyes. The reason is that a fish's stomach is roughly the same size as its eye.

I should warn you that the shop advice was not the best. Allowing the water "to settle" for 6 weeks did nothing except let it sit there. If you had added ammonia during that 6 weeks, or even fish food, that would have been much better advice. As it is, when you added the fish it was exactly the same as adding them the same day you set the tank up.
Telling you to change the water 2 weeks after adding the fish was also bad advice as they could well have died from ammonia poisoning in that time. It is standard practice to change 50% daily if a fishless cycle with ammonia has not been done before getting fish.
The cloudy water could well be due to overfeeding. Have you been cleaning the bottom of the tank? If you have gravel, you need to push the siphon tube right down into it to suck out any uneaten food and fish poo. These will decompose releasing organic chemicals into the water, and there are bacteria that feed on these chemicals. They multiply very rapidly and we see them as white cloudiness in the water. Cleaning the gravel, water changes and cutting down on the feeding will help sort this out. Once their food supply is consumed, the bacteria die and the water clears.

Algae is caused by too much light (how long is your light on for?), by ammonia in the water and worse, the two together.
You have quite likely had ammonia in the tank water, caused by the bad advice from the shop. After the 8 weeks (or whatever it is now) since you got the fish, you should have grown enough ammonia eating bacteria in the filter to remove all the ammonia made by the fish, but you may not have grown enough nitrite eating bacteria yet. Do you have testers for these two, ammonia and nitrite? If you don't, can I suggest getting a sample of tank water tested for them, and get the shop to write down the actual numbers; words like fine or a bit high aren't very much use. The only safe level for both is zero. If either, or both, is not zero, you need to do daily water changes to get them to zero.

I know the neons have survived so far, but prolonged exposure to ammonia and nitrite does lower their immune systems and make fish more prone to disease; and it shortens their life spans. Keeping the water perfect from now on will help them live longer and healthier.

Lights:
12 hours is a long time and with no live plants, you don't actually need to have lights. However, that would mean you wouldn't be able to see the fish properly so I would only have the lights on when you are in the same room to enjoy them. 4 or 5 hours would be plenty for an unplanted tank. I use timers on all my tanks, means i don't forget to turn them on or off.

If you ever do get live plants, then I would be advised by people who know more about plants than me for how long the lights should be on.

Food:
I have a lot of small fish, that's why I crush flakes up, and even grind big pellets to much smaller ones. That way you can feed a small amount and most of it gets eaten.

Gravel:
Leave the fish in the tank, it is less stressful to them than catching them.
The first time you clean the gravel, you may find so much muck that you can't get round it all at one water change. Start at one corner, push the siphon tube right into the gravel, wiggling it gently, and watch till no more muck goes up the tube. Lift the tube out, push it into the next bit of gravel, and so on till you've removed as much water as you want. If you haven't done the whole floor, start next time where you left off.
Once you've done everywhere once, it won't take as long next time. Just make sure you clean some gravel at every water change. The amount of muck down there will surprise you. I have sand in all my tanks and the muck sits on top - it can look very messy if I don't do regular water changes.

Just to check, you do have something along these lines to clean the gravel? Basically a wide see though plastic tube, a blue connector and a length of tubing. The wide end goes into the gravel, and the gravel swirls round inside it then drops out when you lift it up but the muck goes down the tubing. Appologies if you already have one 

That type of gravel cleaner comes in two types: with a non-return valve in the blue bit, or without. The 'with' ones are easier, but take a bit of getting used to. I would advise you to practice with a bucket of water before you use it in the tank. You hold the wide platstic tube vertically, immerse it completely and shake it up and down quite rapidly until the water starts to flow down the narrow tubing. Once you've done it a couple of times it'll be easy. But the first time, you'll get water everywhere which is why I suggested practising first  In the garden with the bucket of water on a box to raise it up is the safest place