Okay here are my thoughts.
Filtration
As a starting point, can you confirm the filtration system on the tank? I've spotted a filter intake and a spraybar, but is is a trickle filter system in the back of the hood area?
What filter medium do you have in there currently?
I would recommend using Purigen as your tank is still maturing and will have some ammonia/nitrite/nitrate spikes - the Purigen will help to smooth those. And also some phosphate remover such as Rowaphos already mentioned, or Chemi-Pure Elite (popular in the US in combination with Purigen for nanos).
Place it somewhere it will get flow (which is also why I'd like to see your filter system).
If you were to spend a bit of money to upgrade the tank, I would look to get an AquaClear70 HOB filter to replace the trickle filter and spray bar, then you can put your filter media in there, even convert to a little refugium to grow macro algae (plus it will look better). They are pricey, around $60-70 from memory, but worth it. In time you can also upgrade to an improved media basket (which cost about the same again ... but again worthwhile in time).
Water
Secondly on water changes - you can't really change 'too much' in a nano (
you could argue the same for larger systems), so don't worry about the amount you change. In fact I would recommend the next couple doing large changes - if you use 25L drums then use a whole one, which would be 80% or so. As long as the temperature is similar to the tank water it will be fine. Ongoing I would increase the amount of the change from 15% up to 25% at least, the more the better really.
Which brings me on to the other aspects of water quality you have discussed. Forget about Alk, Calc and Magnesium at the moment. The NSW and water changes you do will replenish these more than sufficiently for the time being. Once you are fully stocked then maybe start worrying about those, but forget them for now, concentrate on the other aspects.
Algae
Now the tough one. After doing all of the above, your next step should be to introduce a more robust CUC. Get a few turbo snails, some cerith/nass snails (both of these live in and clean the substrate), maybe a hermit or two depending on your view of these (personally I don't like them!). Also see if you can get hold of some other things good for the tank - some bristle worms, stomatellas, mysid shrimp, amphipods, copepods ... all of these help to keep your tank clean.
Is the algae on the main rock? If the rock can be removed I would look into Hydrogen Peroxide applications (I can give you more details if this is the case) ... but if it can't it is a bit trickier, and you'll more so rely on your CUC and approaches above.
Next steps
I would not buy a new tank and cycle some new rock ... you'll likely just get algae transfer onto that and be back to square one, at which case you will probably give up on it all and sell your tank! Stick with it for a while longer at least, might take another month or two to rid the tank of algae (or at least reduce it to a manageable amount) but it's better than starting again. Unless you get really sick of it and lose motivation, then maybe consider that approach.
Overall don't feel like you are failing - algae issues occur in most new tanks, particularly the first on you start
If any of this doesn't make sense let me know.
Cheers