Purple Algae?

Purple algae is just showing off!

Hi Jesnon,
“Getting serious”  now  – you have BBA (Black Brush Algae) appearing in your tank. This algae does not turn up until later on in a tank so it is a no surprise that you did not have any before but now have some... Moving did not help either but believe me, BBA would have surfaced up anyway...
For the time being, what you can do: take the plastic plant out,  pour some bleach over the BBA, leave for 5 mins or as long as you like. You will see the change of colour as the BBA is dying.  Dead BBA is much easier to scrape off (It is quite tough!). I would then rinse the ornament under tap for a good while – and then put it in a bucket with a good overdose of declorinator (10 times), “wiggling” the plastic plant a few times in this and leaving it for a while there before you return it into the tank. This is me – I don’t like taking any chances...
There must be a cause for the BBA in your tank and it will reappear... It depends where it settles onto – you can actually leave it to be if you like (many people with just a few plants or non planted tanks do) – or think about eliminating the cause for it...
As for the brown algae – you probably know that this type of algae is typical for new set ups. As you moved, you had no choice but to disturb the balance in your tank. It should go away eventually (rubbing it off before water changes can help with it at the moment). If it stays for long or continues to re-appear, then it will be a bit of a worry – that would mean you have a “new tank syndrome” continuing for some reason which will have to be really dealt with seriously (but no reason to worry at the time being )...
P.S. Small white balls – this is something not usual for snail eggs. “Pesky” snails tend to leave their eggs encased in a jelly-like “strip” or “blob”, Nerites (I cannot remember if you have any) leave theirs stuck to surfaces (hard to remove and looking like sesame seeds), but “white balls”? Can you take a photograph?