I know there are some old packs with 24x resolution, like this, but all new packs are either 16x, 32x, 64x 128x or 256x...
So, does Minecraft still support 24x or 48x resource packs?
Yes and no. Having even a non-power-of-two texture will disable some or all of the mipmapping options. Basically it'll try to find a common multiple that is a power of two (24x will restrict MC to an 8x mipmap texture, for example), but if there's an odd number or a number that doesn't have a power-of-two common denominator mipmapping will just be disabled while using that pack.
It may also cause problem with stitching on some graphics cards that don't handle non-power-of-two resolutions well. This can cause the sky to show through in a grid-like pattern while in game for some people.
Obviously both of these problems make the game look worse and not run as well, so be careful when doing that.
... excuse my ignorance, what are the effects of disabled/not-well-working mipmaps?
I suggest reading up on what mipmapping is. Wikipedia has a passable article for you to read, so I'm not going to explain the full thing here.
The sort version is that it's a way for Minecraft to lessen it's graphics load by rendering smaller versions of the textures at far-away distances. Because let's face it, you don't need to be able to see absolutely every pixel in a texture when it's off on a mountain in the distance. So instead of wasting graphics power on that far-away stone texture, mipmapping uses a lower-resolution version instead. Like if you start with 32x, you'll see that up close. Once you're far enough away, it'll display a procedural-generated 16x version of the same pack. Further away still it displays an 8x version and so forth. But you can't tell because the shrinking block on the screen loses detail anyway so you usually don't notice... or don't notice much.
When the textures can't be conveniently resized to a quarter of their normal size, now Minecraft has to do some gymnastics to find a common denominator that does work... or else just disable mipmapping altogether. Either one of these slows the game somewhat as it either has fewer cost-saving resolutions to work with, or else just can't do it at all. Either way performance will suffer... though how much depends on the system. The game will also look grainier at a distance instead of the smoother versions that you see when mipmapping is turned on. (You can easily compare them in-game if you want since you can disable mipmapping.)
lol I always played with Mipmap Levels: 4 and Mipmap Type: Nearest - rough smoothing
Well, if I make block textures 16x and item textures 24x I should be fine... right?
It also keeps a cozy look if blocks stay 16x
Yes, that will work. Mipmapping only really matters on blocks and the nearest common denominator is 8x for both so I don't think you'll lose mipmap levels. Minecraft will probably still complain about it, though.
Is there a way to not mipmap signs but mipmap everything else?
Signs in the world are rendered as entities, so I'm pretty sure they don't get mipmapping no matter what you do. As long as the item form is still a power of two you should be OK to change signs as you please.