Have you tried to purposely grow algae?

Nope, I seem to end up with enough of it unintentionally anyway =D

I have but sadly it seems like any tank of water I leave outside gets bugs that start to grow in it and it becomes pretty nasty.

Yes, normally try to grow out some green algae to use as food for fry or to allow for other fish/invertebrates to get a nice food source.

Yes I have, I slide the light to the front or back of the tank so it is shining brightly on the glass wall and in about a week there will be a nice coating of algae that flows with the current that I see some fish pecking some bits off.

Some types of algae (eg. spirulina) is considered a Super Food for humans packed with nutrients so I would assume some would be good for the animals who evolved to graze it

I've never done this. I honestly have never even considered it! I suppose it would make sense for some. I imagine that I would grow it separately like on rocks as to keep the algae in somewhat control.

Is it good for the fish and aquarium to grow algae naturally on it?

Yes, I have purposely attempted to grow Algae. I own Sea Monkeys, hybrid brine shrimp, and they feed off of Algae, so it is a complete necessity for me to grow it.

I haven't been in need do grow algae on my own because they are always of the habit of even over growing, so I focus more on trying to constantly control them not to over grow in my tank.

I don't even have to try, algae grows all on its own in my tank. I can't say growing algae is healthy for your fishes because there is no way to control the algae's population.

No I have not tried. I always had to spend time and energy to remove it from my tank.