This is probably a rather stupid question. I currently ride a 61cm frame, would I see a noticeable difference if I got a 60cm or 62cm bike?

Yes or no. Forget that frames even have a size. Measure top tube lengths and compare. Given the wide variation in how frames are sized, you may find that a 60cm from one brand will be too big and a 62cm from another is too small. Top tube length is a universal measurement that can actually be compared across brands (make sure to always use the 'virtual' measurement if it is a sloping top tube). If you know the top tube length on your current bike and like it, all you need to do is find another frame within +/-10mm and have at it.

You may also want to consider head tube length depending on how aggressive or relaxed you want the fit to be. Since you own a frame already you have a decent baseline from which to compare other frames.

You can adjust for that small of a difference with a stem change. So I'd say probably not.

All else being equal, no difference.
I ride frames between 61 and 68, as long as the top tube is at least 61 cm, it is just a stem swap difference. I have yet to see a head tube that was too tall to achieve a very aggressive position in that size range.

1cm is a small difference, I've ridden everything from a 58 to a 64.

Generally I find that the shorter top tube, longer stem combination will keep my contact points the same distance from each other but rotate my weight into a slightly more aggressive position over the front wheel and allow me to go lower. That makes for a better-handling bike, I think. Go up in size and shorten the stem if you want a rather more upright, relaxed posture.

The only problem with this advice is that reach is not only dependent upon top tube but also upon seat tube angle and head tube height. For every difference of 1 degree in STA, you have to adjust the required top tube length by approximately 1 cm to get the same reach. So you would need a 1 cm shorter top tube on a bike with a 74 deg STA to have the same reach as a bike with a 73 deg STA. That is because the saddle position will be about 1 cm further set back on the steeper seat tube/seat post when properly adjusted to be the SAME distance behind the bottom bracket.

Similar things happen with the height of the head tube.

Overall, your best bet is to compare the more independent stack and reach frame measurements which are not a standard part of the geometry tables supplied for every frame. Depending on how old your current frame is, you may be able to find that data for it as well.

As said above, however, the differences you are talking about should be manageable with a stem change or no change at all.

First of all, you need to realize that geometry of a frame cannot be stated in one number. If you can, compare at least 2 numbers: stack and reach. Comparing more numbers will help you figure out what fits you better.