Are there known necessary and sufficient conditions that specify in terms of an algorithm in a real arithmetic model (where real operations, elementary functions, and comparisons are elementary steps) when a first order differential equation $x'(t)=F(x(t),t)$ with a real-valued rational function $F(x,t)$ in real scalars $x$ and $t$ is exactly solvable by elementary functions and finitely many integrations? 


Useful (positive and negative) partial results? In particular, are there strong sufficient conditions for rational functions with low numerator and denominator degree?



Are all rational exactly solvable differential equations known?
I don't think a general answer to your question is known. I personnaly doubt that it could be positive. 


1. Partial decidable answer: if the variational linear differential system obtained along a given (algebraic) trajectory is not solvable by quadrature (condition expressed as the virtually solvability of its [Picard-Vessiot group](https://mathoverflow.net/questions/140849/solution-of-linear-ode/140851#140851)) then the equation is not solvable by quadrature. This is a theorem by Casale (an improvement on the works by Morales and Ramis) available [here](http://perso.univ-rennes1.fr/guy.casale/research/GC-Malgrange.pdf): 
«Morales-Ramis Theorems via Malgrange pseudogroup, Ann. Institut Fourier 59 n 7 (2009)»


If I remember correctly such a condition can be algorithmtically checked (although you need to address the "equality-to-$0$" problem).
2. Partial undecidability answer: if your real arithmetic cannot decide the equality to $0$, then the Poincaré question is undecidable. The question asks for the existence of a rational first-integral of the differential equation. This is easily checked by considering the linear system $$t\dot x=\lambda x$$ with constant $\lambda$: such a rational first-integral exists only if $\lambda$ is rational (which is not decidable under the assumption). There does exist «elementary closed-form» first-integral, though, *e.g.* $H(t,x)=xt^{-\lambda}$, which are totally acceptable for answering your question (as you point out in the comments). I just thought that mentionning this fact could shed some light on "concrete" diffculties if you try to deal with actual equations.


