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What license are these under? Once I've bought them will I be able to use them in a commercial project? Is edits within that project allowed?

Yes, you can use this tileset in a commercial project. 

You may not host the contents of this zip file\7zip in whole or in part on any publicly avaliable server, unless as part of a game or similar product. 

If you use this material in a game or other product, please give
   attribution to Aleksandr Makarov in the credits.

You are granted the right to modify the artwork to suit your Product.  Remix, transform, and build upon the material for any purpose, even commercially.  

Soon I will supplement this tileset with more elements. I have big plans for it :)

 That's it! Use in any for profit / not for profit product, and give  me a shout out at @IknowKingRabbit :)

Thanks for the quick reply. :)

:D I can even faster, but work + time zones. 

Does this license apply specifically to the person you responded to, or everyone who purchases the asset?

(+1)

For everyone who purchases the asset.