First, I can't conceive of a hermit who is not a contemplative and becoming more and more a contemplative every day. It is part of the very definition of the word hermit as far as I understand eremitical life. Some hermits will, therefore, also be mystics, meaning not merely that they have been immediately gifted by God with mystical prayer and therefore, will have grown in their contemplative lives to the deeper or infused forms of contemplative prayer; it also means they will have had their hearts remade entirely in terms of the virtues and God's love; a mystic is the dwelling place or tabernacle of the active and creative Mystery or depth dimension of all reality whom we call God; they live in greater or lesser degrees of union with God. Such union with absolute Mystery which only God can bring about is evident in their prayer but also in their ordinary lives, and so we call them mystics. While such persons may suffer as all human beings suffer, and sometimes quite intensely in their currently unrealizable yearning for final or ultimate union with God, I don't think any mystic would liken coming to greater degrees of union with God --- the very thing we are made for and come to joyful rest in --- to an affliction like cerebral palsy or autism.
We are all capable of becoming mystics -- even though God alone empowers the deeper expressions of contemplative prayer. It's quite a paradox!!! In fact, as just noted we are all "made for" this degree of prayer and life in union with God who is, again, absolute Mystery --- though few will experience it in their lives. In the Eastern Church the process of growth toward mystical prayer and union with God referred to here is called deification. Few "achieve" it this side of death, unfortunately, but all are made for it. To that extent I believe we can say we are all born to be mystics (those who experience union with God that is wholly God's immediate gift), but I don't think it is appropriate to say some are born mystics and others are not. Moreover, simply because one has occasional mystical experiences I believe the use of the term mystic is still to be cautiously applied. Mystics are not primarily about mystical experiences or phenomena like visions, locutions, and the like; they are first and last about union with God and that means these persons are shot through with Divine love and are transparent to it in a way which, in Christ, makes them into the very imago Dei they were made to be --- whether they are in prayer or living their ordinary lives. If one can say a person's life is defined by (i.e., conformed to and transformed by) immediate experiences of the Love which is God's very self, then I think we can say the person is a mystic, no matter the attendant and secondary phenomena.
Personally, I still prefer the term contemplative, in part because it is easier for folks to understand, but also because I am a contemplative who has occasional mystical experiences (that is, immediate experiences of the God Who is Love) as a kind of subset of this larger category of prayer; Moreover, I look towards union with God as a goal I am called to by God himself, not as a kind of achievement I want or need to point to. I don't think most parishes would have a problem with someone having occasional mystical experiences during liturgy, for instance, so long as the community understands what is happening. Ordinarily, the person praying in this way is profoundly quiet; this may even mean that one's breathing might cease or become indiscernible. Thus, unless one has explained the situation to others they might be concerned about a medical emergency, but if the situation was occasional and understood I don't see where it would be a problem.
Other manifestations need not, but might well be or become problematical, and that would include the ability to read others' hearts. While one might have this ability, one does not need to reveal it to people, and prudence says ordinarily one should not do so apart from a strong pastoral need and authority. My sense is that God would only gift someone with such an ability in instances of exceptional need, along with the capacity for profound compassion, humility, and discretion, not to mention more than a modicum of tact. Ordinarily, this gift is associated with confessors who read into the heart or conscience and assisted the person in moving toward greater union with God. Outside of anecdotes about a number of priests I am not sure I know precisely all this gift entails, but I do know people who are profoundly perceptive about people and seem to miss nothing of what is going on with them. One of these is marked by her compassion and discretion as well and this may mean she can see into others' hearts in ways most of us cannot. When the two qualities are combined there is nothing scary about it --- though it can be unnerving initially until it is clear the person never judges others and does not otherwise misuse what they see/know. Still, perhaps this is only a weak approximation of the gift some confessors have been known to have.
Does the Church have room for mystics? Of course! In fact, she needs them and has always done --- even when their presence has been challenging or hard to deal with. But I think she has even less tolerance for those who are not genuine mystics, meaning those who might want to be recognized for "gifts" without being deeply prayerful, profoundly loving, and practiced in the virtues. In other words, mystics have a certain degree of holiness and that can be/is inspiring to others in ways we all need. What we don't need, however, are those who drop into what is supposed to be ecstasy-on-schedule or trance-via-trigger (I am thinking self-hypnosis here), or those who pretend to have been given the gift of reading others' hearts while demonstrating not the least bit of compassion for those others or true insight into themselves.
Once again, as I have written before, "by their fruits, you shall know them"; the primary measure of the true mystic will always be their capacity to love as Christ loves, to be virtuous as Christ is virtuous, to be imago Dei or imago Christi as every person is ultimately called to be ("I, yet not I, but Christ in me!" is one of Paul's ways of describing himself as Christian and mystic who has experienced a degree of union with God). Only secondarily is such a person's life/prayer marked by mystical phenomena and I sincerely believe it is unlikely in the ordinary course of things, that such phenomena would be known to a larger parish congregation.