A new plant won't harm the cycle.
Of the things you mention in your first post -
There are two types of cloudy - green (an algae bloom) and white (a bacteria bloom). Bacteria blooms are common in new tanks. The bacteria which cause the bloom are not the ones we want to grow, but they are harmless to fish. These bacteria live floating in the water and they eat carbon based food, unlike the filter bacteria which live on surfaces and eat nitrogen based food. The bloom bacteria also multiply very quickly.
If yours is a white bloom, have you been cleaning the bottom of the tank regularly? Good maintenance includes weekly water changes of at least half the water and removing fish poop and uneaten food from the bottom of the tank - and down between the particles if there's gravel on the bottom of the tank. Any poop or food left there will feed the bloom bacteria.
If it's a green bloom, that's caused by microscopic algae in the water. In this case the questions are - how long do you have the tank light on for and do you overfeed the fish so that there's left over food? Both of these will encourage algae to grow. Unless there are live plants in the tank, the lights only need to be on so we can see the fish. Even just 2 or 3 hours a day is enough. And as with bacterial blooms, remove any uneaten food from the bottom of the tank.
On the subject of lights, it's important to have the tank lights on for the same length of time at the same time of day every day. A timer is the easiest way to do this. And the room should be in daylight or the room light be on for at least half an hour before and after the tank lights are on so the fish's eyes have time to adjust going from total dark to bright light and back again.
Algae and rust colour.
The rust colour is probably diatoms, common in new tanks. As the tank matures, it usually goes away. But every tank is different so it's impossible to say how long.
Algae - same causes as algae blooms (see above) only these algae attach themselves to things rather than floating. Probably caused by the tank lights on too long. if you'd had live plants, the fertiliser for them could also contribute but as you don't have any we can ignore fertiliser as a cause.
As I mentioned in my last post, fish shop workers are the last people to take advice from. You aren't the first and you won't be the last to have problems by taking their advice.
To keep the neons and cories 'happy' they need a much larger group of each and a bigger tank. That's something that Pets at Home won't have mentioned. There are really only two alternatives - rehome the neons and cories or buy a bigger tank.
Shoaling fish like these give in groups of hundreds or even thousands in the wild. When there aren't enough of them their instincts tell them that something has eaten the rest of the shoal so they are constantly on the lookout for the predator that may be coming to eat them as well. This is stressful for fish and stressed fish get sick more easily.
But the non-shoaling betta is perfect for your tank