I wanted to ask if it is possible to play on a Minecraft server similar to e.g. Bastighg in his challenge videos?

Since I often watch the videos of Bastighg, Fabo etc. (e.g. Challenge videos) I asked myself whether there's a similar server on which you can start a timer or do a challenge with a friend etc.

I look forward to an answer!

So far, I'm not aware of any server that offers this. You can create your own? Thats is quite easy. You can then also download the plugins. Websites like http://www.aternos.org offer free servers.