First off, I never would've expected to see a singular response, let alone get this many answers and viable tips and information. So needless to say, I couldn't be more thankful. The reason I didn't specify my location, currency etc was because I thought it'd take up more of your time and make things more difficult, yet not doing so did just that to begin with so my apologies for that. Anyways, most options listed above are indeed out of the question, obviously. Now to leave no blank spaces, I'll quickly cover a few of these questions that still arose before my final thought.
Glad to help - but, knowing the location can make things easier, sometimes. Either some of us will be able to help, or we won't, but at least we'll have a more solid answer as to what the limitations are. Don't feel like you're being too much trouble.
A tiny 1366 x 768 screen.
In a way, this makes things a little easier, as, a lower-powered graphics card such as (in order of higher to lower performance, though they are fairly close to each other) a GTX 1050Ti, an RX 560, or a GTX 1050, can perform at your monitor's resolution as fast as, what an RX 580 or GTX 1060 would be required to manage the same speeds at 1920x1080.
(also, if you come across references of an RX 560-8926vs an RX560-1024, go for the 1024 variant).
HOWEVER - depending on who makes the motherboard, sometimes older motherboards won't work with newer video cards, so that's something to consider (although, you usually see that in OEM systems like Dell and HP, rather than motherboards that are purchased as part of a custom build, such as your ASUS board - still, it's a possibility that a newer card might not work).