On AH there are so many djs, there would be someone who could let us know what they do, if they ask permission to someone (like told AndyAce), or if they buy tracks on a specific site which allow you to use them in a djset,....
the only sites with that will let you do that is sites that offer "free" music (not free as in beer, but free as in speech - don't ask me for any names though). but you won't find any tracks signed to any regular labels on such sites, i.e. nothing that would be available from beatport for example.
most bedroom djs won't bother with asking any permission, and labels usually don't care much about those djs posting sets either. i certainly don't have permission to post any sets, but i still do. that doesn't really make it legal though...
the answer you posted from Beatport is aimed at live gigs and playing in public
maybe, but it's still true for all other aspects as well. when you buy a cd or a record, you cannot freely give out copies of it. the same applies to mp3s you buy. and since dj sets consist of tracks, posting a set for download is essentially the same as giving away copies of all the contained tracks.
at least that's the legal theory. in practice, the copyright holders usually won't care much because 1) the tracks aren't contained in original, full quality (they're mixed) and so aren't easily usuable for other djs, and 2) the number of downloads is usually very limited. however as i said before, that doesn't make it any more legal.
a podcast is just an mp3 download, right? in order to have a fully legal podcast, you have to purchase a license (similar to a radio broadcasting license). the license entitles you to give out a certain number of copies, let's say 100. this means you have to monitor the downloads, and when 100 downloads have happened, you'd either have to pull the podcast offline, or purchase a bigger license. i'm sure that at least some of the bigger names have a "legal" podcast like that, because they're a big target for copyright issues.
... but its very hard to control and regulate what DJ's make available for download on the internet because people from all over the world (different regions with their own rules) will download them, so than which rules apply?
the rules of whoever makes the upload available apply.
some P2P users say to defend themselves: "it's not illegal to download stuff, it's only illegal to upload stuff." and that is actually partly true, because it's the person who makes a download available who has the control over who downloads it and how many times. and it's them who has to make sure that the download is fully legal.
(think again of a cd you bought. imagine you put it in a public place, together with a pc and a stack of blank cds so everyone can copy it. then you put up a sign "copy this cd, it's free". if you do that, you're the one to blame, not the people who actually make the copies and take them. of course in practice people would rather take the pc with them than just a cd...)
if you're from italy and you want to have a legal dj set download, you have to go to the organization that handles royalties in italy and purchase a license for a podcast, potentially a seperate international license. (for example in germany this is handled by GEMA, in austria it's AKM and possibly IFPI, in britain it might be PRS, etc etc.)
usually buying a track entitles you to give out a limited number of copies to family or close friends (depends on your country's legislation, def. not the case in the US). however a public download doesn't classify as a limited number of copies under this clause...