I find the hills in general a bit too feminine and a bit distracting. We want to move away from the association with foothills... if the hills can be integrated so they aren't so front and center that would be best.
Entry #4 has the nicest colour scheme.
The font usage in entry #7 an #6 seem to bridge the F&R instead of the "R&I". I want it to read as "f - R.I." not "FRI" or "FR - I".
I do like how you keep the lowercase "f" but I wouldn't mind seeing some different fonts. Something closer to our original fonts (Bodoni MT for foothills and I think Garamond for research institute) could be used for the lettermark and the tagline could use the fonts you have chosen.
We really want to focus on "fRI" lettermark and drop our full legal name from the logo. A combination of entry #1 and #4.