Are there rogue planets between Sun and Proxima Centauri? Would we be able to detect (via emitted radiation, or its gravity) a rogue planet between us and Proxima Centauri? How big would that planet need to be?
 A: Transit-based methods and astrometry won't help here (a rogue (not rouge) planet is highly unlikely to move across the line-of-sight between us and neighboring stars more than once), but microlensing events like that (provided there's another star behind the planet and Proxima) are possible.
Of course, if you don't make repeated observations, you cannot be sure how far the planet is, what its proper velocity vector is, and whether you are detecting a regular planet or a rogue one.
In principle, Earth-sized planets are currently detectable via microlensing.
When James Webb Space Telescope is launched, it may be able to spot and possibly track a rogue planet with a still liquid core, heated above the background temperature.
EDIT: I must admit I've fallen behind the times. WISE_0855-0714 with mass 3 to 10 times that of Jupiter was discovered 7.53 light years away from us by Kevin Luhman with the WISE space-based infrared telescope. This is beyond Proxima Centauri, but certainly in our closest neighborhood. The temperature is quite high (−48 to −13 °C or -55 to -8 °F)
 
Source: NASA via Wikicommons.
