No you don't need to post proof of participation anywhere. Just meet all the requirements, complete the form and everything is good

I did everything, completed the form one minute after the start, obviously did some mistake since I didn't recieve any coins.
Am I allowed to apply for the next airdrop?
The most common mistake was people not pasting the correct link for their Bitcointalk profile. Make sure you're link has numbers in it. You can find it by clicking your profile name in a thread like this, or by going to profile and clicking show all posts and copying that url.