do not lose much time reading this unclear answer, i will fix it, if god wills / insha'a allah . ( pedophilia term has some proofs. dislike of it is felt. some things which are understandable by people are not said in quran and hadithes. )
1) yes, islam supports child marriages; muhammad's case is not exceptional. 2) what is pedophilia? look at wikipedia: "an adult or older adolescent experiences a primary or exclusive sexual attraction to prepubescent children". so, does islam support sexual attraction to prepubescent? there are some hadiths and ayats that support desire to marry virgins: for example: http://sunnah.com/search/?q=jabir+marry :
He said: 'Why not a virgin, so you could play with her and she could play with you?'"
http://www.islamawakened.com/quran/55/74/default.htm (about paradise) :
Virgins, in a state of inviolate chastity, whom no man or Jinn has ever touched,
but desire to marry is not strictly equal to sexual desire. desire to marry may be with sexual attraction to future state of body, not to current state of body. additionally, girls should be married for her religion, not for beauty or other things, according to hadiths http://sunnah.com/search/?q=marry+four+dust .
i think, sexual intercourse with girl who cannot conceive is israf, and israf is prohibited in islam. but in other hand, it may be accounted as act that makes spouses closer, so, i am not sure, and islamic scholars allow it. (of course, it is prohibited if it is painful). and maybe it is even not possible to know out when first ovulation occurs, so it is has some additional logic in that case.
so, i think, islam does not encourage nor discourage sexual attraction to prepubescent, but, encourages child marriages (due to it encourages marriages generally without direct age restriction), and it is controlled by spouses an their families, what and when they do. and marriage does not mean primary or exclusive sexual attraction to prepubescent, because marriages in islam are done without time terms, and divorce is discouraged.
if you/they ask, why islam does not have direct marriage age limit/restriction or why it does not discourage sexual attraction to prepubescent, answer is - why it should ? and the european tradition also does not discorage all sexual attraction to prepubescent, as you see, only primary or exclusive of it is discouraged (in new edition of wikipedia it is said : "Pedophilia or paedophilia is a psychiatric disorder in which an adult or older adolescent experiences a primary or ..." ).
and take in account, that even if this word/term "pedophilia" looks/sounds medical and scientific, i have not seen any scientific base of it, so, it is just an european culture element, and it has not to be followed as some scientific fact. by this definition given in wikipedia it is indeed a disorder, but, it, as it is defined in wikipedia, probably does not have any proofs anyway, and when i looked for sources of this definition i see that it is not as it is in sources! source no 1 is DSM-5 and it is not accessible but definition of it is cited in wikipedia:
... pedophilic disorder ... paraphilia in which adults or adolescents
16 years of age or older have intense and recurrent sexual urges
towards and fantasies about prepubescent children that they have
either acted on or which cause them distress or interpersonal
-- so no "primary or exclusive" is here!, source no 2 is ICD-10, definition of it :
F65.4 Paedophilia A. The general criteria for F65 Disorders of sexual
preference must be met. B. A persistent or a predominant preference
for sexual activity with a prepubescent child or children. C. The
person is at least 16 years old and at least five years older than the
child or children in B.
F65 DISORDERS OF SEXUAL PREFERENCE G1. Recurrent intense sexual urges
and fantasies involving unusual objects or activities. G2. Acts on the
urges or is markedly distressed by them. G3. The preference has been
present for at least six months.
-- so neither, no "primary or exclusive" is here! and this contradicts rules of wikipedia and somebody should fix the definition in wikipedia. i think, adding "primary or exclusive" is just a trick of editors of wikipedia to make the the term correct , because almost any attraction, if you add "primary or exclusive" to it, becomes a strange behavior, for example: attraction to drink is normal, attraction to tea is also normal, and we cannot understand, why it would be a disorder, but if write "primary or exclusive attraction to tea", we understand that it is a starnge behavior, because there is no reason to want to drink only tea.
and what i can say about these definitions? these does not say about just sexual attraction, as it is written in wikipedia, but about "sexual urge" and "preference for sexual activity with ... f65 g2 ... urges". as the word "urge" is used, it is not just any sexual activity like hugging but about sexual intercourse. and sexual intercourse with prepubescent child spouse is indirectly discouraged by islam, because of israf is prohibited, as i have already written. and these definitions also does not have any scientific proofs, explaining why it is a bad thing. and if they would try, i think, they could not proof it, (with exactly these definitions), because there are different people, and even sexual intercourse with little child maybe physically harmless, for example, if it is non-penetrative. but non-penetrative sexual intercourse is discouraged in islam: http://sunnah.com/bukhari/64/182 : "... coitus interruptus ... Allah's Messenger (ﷺ) ... said ... It is better for you not to do so ...", also it is a discouraged thing because it is israf. and that child must be spouse in islam. it is psychologically harmful and bad thing if it is without marriage and it is disallowed in islam. and, i think, marriage with too little child is indirectly prohibited in islam, because he/she cannot give proper consent for marriage.