Why is Iron very rare in seawater compared to many other elements? I don't know the associated technical words to research this question. 

There was high presence of Iron in the oceans in anoxic precambrian times, and today copper is 5 times more abundant in the oceans than iron. 

Is the solubility of $\ce{Fe^{2+}}$ and $\ce{Fe^{3+}}$ and $\ce{Cu^{2+}}$ very different? 

Why is there five times more Copper in seawater than Iron? Surface minerals have 1000 times higher PPM values for the latter? It's precipitating?

Why is there 4000 times less iron in seawater than Strontium? Surface minerals have 100 times higher PPM values for the iron? 

Here are some measurements stated previously: 
https://web.stanford.edu/group/Urchin/mineral.html
https://en.wikipedia.org/wiki/Abundance_of_elements_in_Earth%27s_crust
I'll elaborate this answer as I learn. 

It's an effect owed to the low solubility of iron in the modern oxic ocean and the tendency for iron to be ‘scavenged’ from the water column by sinking particles.

Brand, L. E., Sunda, W. G. &amp; Guillard, R. R. L. Limitation of marine
phytoplankton reproductive rates by zinc, manganese and iron. Limnol.
Oceanogr. 28, 1182–1198 (1983).

Cooper, L. H. N. Iron in the sea and in marine plankton. Proc. R. Soc. Lond. B
118, 419–438 (1935).

Goldberg, E. D. Marine geochemistry. 1. Chemical scavengers of the sea.
J. Geol. 62, 249–265 (1954).