Several things come to mind straight away. If the real Satoshi Nakamoto started this forum in 2009, did the present administrators get hold of it without interacting with Satoshi?
No. There was also Sirius which helped Satoshi immensely with the Bitcoin project in 2009. My knowledge on the history is limited because I was new at the time, but I believe they both trusted theymos to take care of it in 2010. Sirius gave control of the domains to Cobra. theymos and Cyrus run the forum now. That link appears to be a bit behind the times as BadBear isn't an administrator anymore, he doesn't appear to be even a moderator. He last posted in late 2016 and his account was last active on here in spring 2017.
If they had direct dealings with Satoshi then maybe Satoshi can be identified. And how did these present administrators get hold of the forum?
Satoshi left the project over 7 years ago and there's no point in identifying him anymore. Sirius, who was responsible for porting the code to Linux and getting a lot of early interest (even more than Satoshi) eventually decided to leave the project in the hands of those who were interested in taking it further. The forum grew a lot from 2010 to 2013 and even though the signal to noise ratio hasn't always been the best, theymos and Cyrus have done a great job given the limitations of the software. theymos also runs the subreddit which has grown quite well over the years.
Was donated to them or did they buy it? Anyone who knows please post here.
It wasn't donated or purchased. Responsibility was transferred to them. Responsibility is NOT a donation, it comes with requirements. One doesn't simply "purchase" responsibility.
Wow... this forum is very lucrative for the present administrators. They have received thousands of bitcoins in donations!
A significant portion of forum funding was used for
development of a new forum. At present, the donation address only has 162.64 BTC. Inflows of ad revenue I believe mostly pay the moderators. It's not much but it's better than nothing. Also, there's likely to be future developments and moderation required, and it is important that there is enough funding for decades (if not centuries) of perpetuation of the community, mods, admins, etc. "lucrative" is a loaded word in the context you used.