The other day I heard a friend of mine speaking Russian to her mom on the phone, and she said, "да, она очень найс", switching to English (nice) for the last word. When I asked her about it, she said that по-русски she would have said <<хорошая>> but that this wouldn't necessarily refer to her disposition specifically like "nice" does, and is more cumbersome. So two questions: 1) is it becoming colloquially acceptable to use "nice" in Russian? And 2) are there other situations like this where a borrowing is coming to replace a Russian word in certain situations (due to increased specificity)?
Well, it's definitely a slang nowhere even near gaining any traction enough to claim it is a widely-accepted colloquial word. Yes, true, some younger (ha-ha) hipsters use it, but I'd rather claim that some of soviet hippies slang was more widely adopted. I'm mentioning hippies mainly because their slang also was heavily influenced by English, like, "хайратый" (from "hair"), герла (from "girl"), шузы (shoes) etc.
I'm not trying to neglect any kind of foreign influence, this happens a lot in any language, it's just that "найс" de-facto is not there yet, if it will be at all.
The same social group uses such borrowings as "харассить", "абьюзить", "фан" (from "fun") - but still, time will tell whether this hype will change the language.
As of the second part of your question - that happened a lot throughout the history of Russian. Some words, like "серьёзный", "юзер", "киллер" co-exist in Russian alongside with their "pure" counterparts.
Is it becoming colloquially acceptable to use "nice" in Russian
No, as far as I can tell, it's not.
This word would probably be not understood by Russian speakers not speaking English and living outside English speaking environment (that is majority or the Russian speakers).
Direct borrowing of adjectives (without adapting them to Russian paradigm) is not characteristic for Russian at all.
There are lots of words in Russian which can be used to convey the meaning of "nice person": милый, любезный, приятный etc., each of them having its own specific.
Since both your friend and her mother are apparently bilingual in both English and Russian, your friend used the first suitable word in either language to cross her mind (most probably because she met the woman she was talking about in an English-speaking environment) and didn't bother with adapting it for Russian.
Are there other situations like this where a borrowing is coming to replace a Russian word in certain situations (due to increased specificity)?
Of course, as soon as there appears a concept worth distinguishing with a separate word, the word emerges in the language, either through borrowing or in any other way.
For instance, Russian for "pastry" is пирожное, a word which was coined in XIX century, first as a collective noun only. A certain type of pastry was at first called английский хлеб, but when it become popular enough to deserve a term on its own, it was named with a loanword кекс (from "cakes"). Both muffins and cupcakes used to be called that, however, later, as they gained popularity in Russia, they came to be known as маффин and капкейк in Russian as well.
However, all verbs and adjectives borrowed from other languages are adapted to Russian paradigms: парковаться, джинсовый etc. If the loanwords are hard to adapt to Russian as adjectives or verbs, they become nouns and are used analytically: цвет хаки "khaki", читать рэп "to rap" etc.