Well, the go-to book is Henry Ott's Electromagnetic Compatibility Engineering, but that's probably overkill for you. You're pretty much a hobbyist that doesn't need FCC certifications.
So is it ok if I provide the case as it is?
The aluminium is punched with some air vents on the rear (4mmx12mm holes), some on the base (same hole dimensions) where the motherboard would sit and some where two 2.5" hard drives would sit (again same hole dimensions).
The emissions from the box are related to the largest holes or arrays of holes. Which in your case are the entirely open sides.
I've heard of nickel based paints that can be sprayed to the plastic parts (on the inside) but is this safe?
Yes, they're as safe as any other paints (i.e. clean up your messes and don't eat them). They've been in use for decades. There are also copper- and silver-based paints. The latter paints are more conductive than nickel; sometimes, you want lower conductivity so the shield acts as a damper.
Does the paint need to be conductive?
Yes. Not conductive is not there. Also, there needs to be a good connection to the other parts of the shield. A wire isn't that great; you really want a metal spring-finger EMI gasket or something similar. The joints are holes in the shield if they're not connected. Again, all the details are in Ott's book.
Is there a risk of short circuiting the components inside, fire or other?
Yes there is, if you have components resting against the case. This is a bad idea in any case design.
If sprayed, would the plastic parts need to be grounded and to what?
Basically, a complete shield works off of the principles of Gauss' Law: A closed conductive surface with a single point charge can represent the same surface with numerous point charges; you just add them up. Put that together with Kirchoff's Voltage Law and connect the conductive surface to your ground, and any charge movement in the surface doesn't matter to the outside world.
Of course, this perfect world falls apart when you put a hole in the closed conductive surface, but then it turns out that for long wavelengths (i.e. low frequencies), if the holes are small enough and spaced out right, it's the same as a closed surface.
Thus you want your shield to minimize the size and number of openings in this conductive shell, and you want to prefer more small holes to fewer bigger holes.
What about shielding tape, is that any good?
It's fine for plugging gaps, but be careful: some tapes have conductive adhesive and some don't. I prefer to avoid tape if possible because you have to put new tape down every time you open that joint. Spring fingers don't have that problem.
Speaking of tape, though: There are insulating tapes (I prefer the Kapton stuff) that can come in handy for insulating spray-on shield rom nearby components. So long as the components aren't so hot as to melt the tape, you're good.
Would I or the user need to cover the plastic parts entirely?
As much as possible. Anything uncovered is a hole in the shield.
What about the bits where aluminium and plastic join?
Spring fingers or clips or something.
And the vents in the plastic parts?
Nothing you can do there that is practical.
And finally how do I calculate the exact area required for the air vents so that it doesn't affect RFI/EMI too much?
With finite element analysis software. To me, that would be a lot more effort than you need.
Overall, I think you can just go build your case as-is and not worry. I've seen unshielded cases made entirely of acrylic (clear plastic), which has no shielding properties whatsoever, for sale in local stores. Also, FCC and other regulatory testing only applies to completely assembled devices, not components that go in the box.