Just ammonia is not terribly likely, for the simple reason that water is so stinkin' common, which in turn is because oxygen is way more common than nitrogen. Just look at the moons of Jupiter and Saturn: there's oodles of water ice there, but no "ammonia moons".
So, if you want a pure-ammonia ocean, you have to come up with some excuse for why water was excluded, which is gonna be kinda tricky to manage. Just freezing it won't work, because water will dissolve into ammonia to form a eutectic mixture; getting it cold enough to freeze out all of the water means you'll freeze all of the ammonia, too!
A water-ammonia mixture, on the other hand, is perfectly plausible, and reasonably common in a certain type of SF. Hal Clement, for example, has several works which assume that life-bearing water-ammonia worlds are in fact more common than straight water worlds like ours, and humans are the odd ones out in the galactic community for that reason.
Ammonia worlds generally, and especially small ammonia worlds, would be more common around smaller, cooler stars, with less UV output. And yes, they would also be more common on the cooler end of the "Goldilocks zone"- especially since ammonia is a greenhouse gas, which shifts the whole zone outwards, and because (as noted in the question) ammonia-water mixtures have lower melting points and wider liquid ranges than pure water, or pure ammonia, so the "Goldilocks zone" for a water-ammonia planet is in fact wider, and extends farther into the cooler end, than for a plain water planet.
Warm ammonia worlds are, however, also still possible, especially around cooler stars with less UV output (in which case they'd need much smaller orbits than ours to end up hotter than Earth is despite the cooler star). And there are a lot of cooler, K-type stars out there! Eventually, though, even with UV-suppression, you run into the problem of hydrogen loss, which becomes a problem for retaining ammonia at lower temperatures than it does for water. Most warm ammonia worlds will thus probably need to be super-Earths, not mini-Earths, with higher escape velocities so that they can retain hydrogen in the upper atmosphere. So, yeah, ammonia worlds could occur in our position in the "Goldilocks zone", and even warmer, but it's easier to form them, and you can get a greater variety, out in the cold.
Meanwhile, the oxidation status of the planet absolutely also matters. A world with an otherwise oxidizing environment will lose its ammonia, and end up with a nitrogen atmosphere instead, while a world with a lot of free hydrogen will generate ammonia, removing hydrogen and nitrogen from the atmosphere. If the world is large enough to hold on to hydrogen against thermal Jeans escape, it'll probably remain reducing, and you can keep your ammonia-water mixture oceans. Life is unlikely to develop oxygen photosynthesis in that sort of situation, since it'll be a lot easier to just grab free, pre-reduced hydrogen from the environment. A world that cannot retain excess hydrogen, but which is cold enough to retain ammonia anyway due to an atmospheric cold-trap such as helps retain water on Earth could be threatened by oxygen photosynthesis... except oxygenic photosynthesis is also unlikely to develop there, since it's a lot easier to pull hydrogen off of ammonia than it is to pull it off of water! As a result, you are likely to end up nitrogenic photosynthesis, which would be a major extinction event like the development of oxygenic photosynthesis was on Earth, but for different reasons--it'll shift the balance between ammonia and atmospheric nitrogen, changing the water/ammonia ratio that organisms have to deal with, but not eliminating ammonia completely.