Does unused thermal paste get old?

I've got two small unlabeled plastic syringes with white cooling paste - these are probably 8 years old. Should I throw them out and buy new?

I would, since a new tube of medium-quality thermal paste costs $5... And not for anything, but I wouldn't put 8 year old thermal paste anywhere near a new computer.

I'll need some paste to build a mid-range system, so I don't want (need?) the best and most expensive cooling paste. If unused cooling paste doesn't deteriorate over time, then I could even use what I've got.

Yes, you could use what you have. It probably still works fine, and as long as it's not dried out and was stored properly (i.e. at room temperature in the dark). It should perform just as well.

That being said, you should highly consider buying new thermal paste, or at least a reasonably priced, decent quality one. I never recommend anyone cheaps out on thermal paste, regardless of how much your system costs. For $5-10, you get a lot better returns on cooling performance, which translates into longevity of your system's components.

The paste itself is likely to still be good for use with the class of processors available at the time you acquired it - however it may not be suitable for more current processors.

Most (though not all) HSF assemblies now come with a 'pad' of thermal interface material pre-applied, so if you're purchasing new components you should be OK even without purchasing additional paste. Check with your supplier and on the manufacturers website(s) to be sure though.

If your chosen HSF doesn't ship with a suitable paste, you would probably be best to buy new - the formulation of the compound has improved over the past few years, it's worth the extra few $/£/€ to get a good interface between your HSF and CPU.

Depends on what type of paste it is. Generally it is safer to replace it when in doubt.

Diamond 7 carat paste is extremely good and not expensive at all, a little difficult to apply (thick, must be warmed up using hot water) but I picked up some the other day at Micro Center and it was cheaper than AS5