Has any research into actually producing anything larger than the F1 been seriously carried out?
The M-1 was a hydrogen engine just a little larger than the F-1. Parts of it were built and tested and the engine would likely have worked just fine if completed and flown. Lack of need for a super-heavy lift vehicle larger than a Saturn V prevented it from being developed further.
The RD-170/171 is comparable to the F-1 -- heavier but more compact due to using 4 combustion chambers instead of 1, and just a bit more powerful.
Although the [Sea Dragon] proposal was never further investigated, would [Sea Dragon's first-stage engine] be possible to fabricate today?
Probably. The design was fairly conservative for all its size, pressure-fed rather than pump-fed, with quite low chamber pressure. As mentioned in the Wikipedia article, the general design was reviewed and considered sound by TRW; I assume that includes the engines.
The main problem I'd expect to see would be combustion instability -- conventional wisdom says that's a bigger problem in large chambers than small ones, and this is certainly a large chamber. This Q/A discusses that problem.
However, the original Aerojet-General proposal suggests that the resonant frequencies of such a large chamber would be so low that feedback instability wouldn't be sustained:
With regard to combustion stability, an analysis on
the basis of sensitive time lag theory (perhaps the best theory so far
developed) indicates that the Sea Dragon thrust chamber will operate
well outside the region of combustion instability. One of the primary
advantages of sea based development testing is that it permits early
experimental evaluation of combustion stability on a full scale basis
without an exorbitant outlay for facilities.
In other words "we don't think it's a problem, but, hey, at least if one of those engines blows up in the ocean it won't hurt anything."