Is the community obligated to support financially those who study Torah all day long? Is one allowed to request such support from the community based on his choice to study full-day and not make a living by an 'ordinary' job?
Rambam is explicit and passionate on this topic (Hilchot Talmud Torah 3:10) . . . NO!!!!:
"Anyone who decides to be engaged in Torah [study] and not to work, and will be supported by Tzedaqa - this person desecrates God's name (Chillel et Hashem), degrades the Torah, extinguishes the light of our faith, brings evil upon himself and forfeits life in Olam haBa (The world to come); since it is forbidden to derive benefit from the words of Torah in this world. The Rabbis said (Avot 4:5): Anyone who derives benefit from the words of Torah in this world, forfeits his life in Olam haBa. They further commanded and said: (Avot 4:5) Do not make them [the words of Torah] a crown to magnify yourself or an axe with which to chop. They further commanded, saying: (Avot 1:10) Love work and despise positions of power (Rabbanut). And: (Avot 2:2) Any Torah which is not accompanied by work will eventually be nullified and will lead to sin. Ultimately, such a person will steal from others."
(Translation from http://www.torah.org/learning/rambam/talmudtorah/tt3.10.html )
Of course, other authorities can disagree with Rambam. But I have yet to see a compelling counter argument. And given the severity of Rambam's case, it should be countered by something particularly compelling.
The answer to this question will depend on your hashkafa, so please take my response as an indication of one individual's tradition, with many other possibilities out there.
No. Communal funds are holy, and may only be used to support the community and other holy endeavours. It is appropriate to support only "the cream of the cream of the cream" (a favourite phrase of my Rosh Yeshivah) with communal funds - those men who will become Rabbis, Dayanim, and Roshei Yeshivah, i.e. directly serving the community. For these people it is not just appropriate but necessary to fund them, as an essential way of building the community and its future.
Your average kollelnik I presume to be a superb learner and he is certainly doing a valuable mitzvah by learning all day, but it is not a proper use of charity money to pay for him when he could be supporting himself. Unless the money invested in him will deliver advantage to the community it should go elsewhere.