Can a solar system exist where the second planet rotates fast, and the third planet is tidally locked to their star? Today I read about the Romulan home star system and it looks like it might be impossible for such a star system to exist.

Memory Alpha describes Remus:


  Remus was tidally locked, with one hemisphere always facing its primary while the other always faced away from it. The civilization of the Remans was located on the night side.


Remus is farther from their star than Romulus. Romulus rotates many times during a single orbit of their star while Remus is tidally locked and rotates once per orbit of their star.

If a planet orbits even deeper in the gravity well of their star than Remus it should be even more likely to be tidally locked than plants further out.  

Could tidal interactions with neighboring planets and/or its moons prevent a planet from becoming tidally locked while being unable to prevent planets further out from being tidally locked?
The tidal locking timescale depends on several factors: $$\tau_{lock} \approx \frac{0.4 \omega_0 a^5 m Q}{3 G M^2 k_2 r^3}$$ such as the initial spin rate $\omega_0$, the semimajor axis $a$, the mass $m$, the solar mass $M$, the radius $r$ and various dissipation parameters $Q$ and $k_2$. 

Two planets that merely differ in $a$ will have the inner one lock much earlier than the outer one. But give them slightly different $\omega_0$, $m$ and especially $r$ and the inner might dissipate more slowly than the outer.

There is also the possibility of spin-orbit resonances. Mercury is in  3:2 resonance, and tidal forces might lock planets into 2:1 or 5:2 resonances. That way the inner planet may be in a resonant lock while the outer is tidally locked 1:1. In fact, Venus is turning very slowly (and retrogradely) so it is not too far off, although the causes of the retrograde spin are unclear. 

So, yes, you can have tidally locked planet orbiting outside non-locked planets.