I've moved the summary to the top, read the whole thing to see my reasoning (or lack there-of) ;)
How small can a creature evolve and become sentient?
Unknown for sure. I would guess about an inch large at the very smallest. More likely is that it requires brains bird-size or larger (simply based on the intelligence of crows).
How likely is this and what environmental factors might lead to their evolution?
Extremely unlikely. An environment which allows time to think and benefits for doing so.
The smaller the creature is, the smaller their brain is. Eventually, it is debatable whether that clump of neurons can really be considered a brain. Worms have several clumps of neurons at different points in their body. Are these brains? They probably don't give enough room for sentience though.
One thing to possibly consider is that anything with a spine (vertebrates) has what we can call an "actual" brain (no offense, non-vertebrates, I know some of you do too). According to Wikipedia,
The smallest vertebrates (and smallest amphibians) known are Paedophryne amauensis frogs from Papua New Guinea, which range in length from 7.0–8.0 millimetres (0.28–0.31 in), and average 7.7 millimetres (0.30 in).
Another source for frogs with similar size
I would imagine that the smallest brain you could get that is sentient would have to be a bit larger than that, maybe about the difference of brain-to-body ratios between humans and frogs. However, this is just my thoughts - I am no biologist.
It could very well be that certain brain functions require a certain number of neurons to pull it off. This is actually very likely, judging by the number of species in this world and how many we know to be sentient. I don't think the answer to that is actually known, but someone can correct me if I am wrong.
There are a lot of factors, in my opinion, why such a creature is not likely to evolve, but here's three:
Looking at the proportion of known sentient species in existence compared to non-sentient species, statistics are simply not in our favor.
The smaller animals get, the faster they need to be able to respond to danger. This is because their perspective is so small. Imagine if anything within distance of how far we could see on a clear day outside could be within range of hurting us. Generally a brain to allow thinking about options and decisions would only slow you down.
Body heat will be an issue. The smallest mammal is the Etruscan shrew or Bumblebee Bat at just over an inch long. These animals have to eat a lot to have the energy to keep their body heat up. Going smaller than them would likely be fatal if your creature is warm-blooded.
That shouldn't stop you from allowing such creatures. If there was a gap in fast-responding predators or the intelligence to create tools aided escape, allowing a great advantage, then it would be believable that it could happen.
