I view these as normal English words, I use them and would not see a person using them as a pedant. On the other hand, I use vituperative too. To be serious, I think alacrity and zeal are not uncommon.
As to the different flavours of meaning, I think the Google definitions actually call out the differences quite well.
Alacrity: the key here is readiness. Typically I would use this to describe how someone starts an activity. They show alacrity at the point of beginning.
Fervour: is about feeling, the inner attitude of the person, their intensity. To me this has negative overtones; English folks don't do fervour!
Zeal: In contrast if somewhat old-fashioned, zeal is positive. It's more outward looking than fervour, how a particular objective is being achieved. I want my team to be zealous. The derived term Zealot is negative, tends to imply a loss of perspective, excessive single-mindedness.
Fanaticism: I'm tempted to say it's Fervent Zeal. Excessive enthusiasm, uncritical following of a path. If Zeal is very much about reaching an objective, fanaticism is more about a broader approach. We don't so much discern the fanatic in the approach to one task but in a general approach to many tasks, both in selection and in the single-minded disregard of side-effects. Curiously, we shorten Fanatic to Fan and it becomes a more favourable concept: "I am a Watford Fan" is taken to mean "I am a keen supporter of Watford".