Is it only about chips and AI for economic reasons or is there more involved like, an army of robots or something to control everything?
There are two main reasons that REE production is dominated by China. One is that Chinese production is low-cost. Two is that the total market demand is small, so that (absent some long term price guarantee) adding another big REE production center outside of China will probably tank the market value and lose the investors money. See the history of the Mountain Pass mine in California:
https://en.wikipedia.org/wiki/Mountain_Pass_Rare_Earth_Mine
This one mine produced most of the world's REE between 1965 and 1995. It closed in 2002 because it had a toxic spill that would be too expensive to remediate given the arrival of Chinese competition. In 2012 it reopened as a reaction to Chinese export restrictions on REE instated 2 years earlier. In 2015 global REE prices went back down and the Mountain Pass mine went bankrupt again. It resumed operating in 2018 after it was brought back out of bankruptcy. It relies on frosty US-China trade relations and government support to remain a going concern. There's not enough global demand for REE to keep Mountain Pass independently profitable whenever Chinese export restrictions get relaxed.
[1] In terms of total market value, I mean. Dysprosium has a high value per kilogram but the current annual demand for all rare earth elements combined is under $8 billion, compared to e.g. $20 billion for zinc alone.
If one views it trough the lens of: What a high school bully, who has no self control and can't delay gradification, would do... then nothing Trump does is surprising.
The reason for the deal is: recover money for taxpayers, provide some security guarantee without an explicit commitment that will end up with US being deployed, and backstop significant private-sector investment into Ukraine to try to revive their economy (after the war destroyed and maimed a huge proportion of their working adult population).
There are (almost) no rare earth minerals though.
I have no idea, so this is pure speculation, but as charge cycles for batteries keep increasing I'm guessing they'll become more valuable, especially if they're made from finite resources. Consider a scenario where you never own the battery and have to lease it forever.
The opposite side would be choking off the supply of the rare minerals needed to make batteries with the intent of making it more difficult to switch to renewable energy. You'd need all of them though and I just don't see them improving their position in Canada.
Or it could be as simple as "gold is shiny" and Trump wants them because he's heard they're valuable.
This appears as very long term thinking, while in reality is a way to siphon money in the short term and sell whatever idea they currently have under this long term plan.
Not to say that it would be bad for US to own Greenland - of course it would be great. Mars too BTW...
Given it's natural ressources and geographical position it's extremely intresting. But of coures the second part is right - it's just not in the realm of possible, without going to an ugly fight with Europe.