Who fasts and who doesn't?
Do, and its important
Both the Eastern Orthdox and Roman Catholic churches have required fasting periods for their members, as shown by the links.
Members of the Church of Jesus Christ of Latter Day Saints are encouraged to fast for the first Sunday of every month, which they call their fast and testimony meeting. Instead of speakers, members come up to the microphone and bear their testimony about the truthfulness of their experiences in the Gospel [per Gilbert LeBlanc]
Encourage, but in practice don't
Episcopalians are supposed to fast on Ash Wednesday, Good Friday, and on "lesser fasts", but I've never heard it preached from the pulpit in the five years I've been one. Pancake Tuesday? Sure. Fasting the next day, not so much.
Baptists practice fasting as a private discipline but have no agreed upon days on which fasting is to occur. (Baptists can't even agree on a lectionary, so if you think you'd be able to pry the fried chicken out of my cold dead hands, think again! When I was ordained as a Baptist minister, I explained to my ordination committee that Baptists only believe in three sacraments - marriage, communion, and potluck. They agreed.)
Methodists encourage fasting but offer no official guidelines on when to do it, since it is a "private matter"
Presbyterians, per the Westminster Confession of faith encourage fasting on Sundays, but do so with the admonition of Matt 6:17, in which believers are encouraged not to brag about how spiritual they are.
Lutherans encourage fasting but do not require it.
Why don't many Protestants fast?
Most Protestants don't fast - not because they think fasting is wrong, but because there is nothing compelling to make them think that it is important. Very few Protestants I know would say fasting is wrong, but in practice, it just isn't important*. (See note on "Papistry" below, however.)
For most evangelicals at least, and many protestants in particular, the emphasis on grace is such that anything that smacks of "works" is relegated to the "nice, but optional" category. Puritans would would call some of the spiritual disciplines "superciliousness" - an over attention to the perfection that leads to pride. Additionally, many older Protestants view fasting as an artifact of Catholicism. This isn't accurate, but for many centuries, anything that smacked of "Papistry" was viewed with suspicion.
All of these basically leads to a culture in which many spiritual disciplines are just not part of the culture. Kids don't grow up thinking you do it, so they don't, and in turn, their kids perpetuate the cycle.
After attending a spiritual disciplines seminar or after reading Foster's Celebration of Discipline or anything by Dallas Willard, they tend to learn, but the point is that the culture of Protestant churches (and I am generalizing a bit there) simply doesn't emphasize it.
Finally, if you ask me, "why don't I fast?" It's really just laziness. I know it would be good, and I have done it from time to time, but in my personal experience it hasn't done a lot. I suspect I'm not alone in thinking that. It doesn't justify it, it just explains it.
Excursis on the textual evidence for fasting
*For example, while I wouldn't argue this, the verse you reference in Mark is considered by many scholars to be a minor variant inserted after the original manuscripts. I won't argue it either way, but it just doesn't rise to "really important" for a lot of Protestants.
From the Net Bible's footnote on Mark 9:29:
38 tc Most witnesses, even early and excellent ones (Ì45vid א2 A C D L W Θ Ψ Ë1,13 33 Ï lat co), have “and fasting” (καὶ νηστείᾳ, kai nhsteia) after “prayer” here. But this seems to be a motivated reading, due to the early church’s emphasis on fasting (TCGNT 85; cf., e.g., 2 Clem. 16:4; Pol. Phil 7:2; Did. 1:3; 7:4). That the most important witnesses (א* B), as well as a few others (0274 2427 k), lack καὶ νηστείᾳ, when a good reason for the omission is difficult to find, argues strongly for the shorter reading.
Matthew 17:21 is often ommitted (NIV, NRSV, ESV, Net Bible, HCSB, for example) for the same reason:
tc Many important mss (א* B Θ 0281 33 579 892* pc e ff1 sys,c sa) do not include 17:21 “But this kind does not go out except by prayer and fasting.” The verse is included in א2 C D L W Ë1,13 Ï lat, but is almost certainly not original. As B. M. Metzger notes, “Since there is no satisfactory reason why the passage, if originally present in Matthew, should have been omitted in a wide variety of witnesses, and since copyists frequently inserted material derived from another Gospel, it appears that most manuscripts have been assimilated to the parallel in Mk 9.29” (TCGNT 35). The present translation follows NA27 in omitting the verse number as well, a procedure also followed by a number of other modern translations.
Just as a translation of some of those symbols, א is the Codex Sinaiticus (aka the Tischendorf's stolen manuscript) and B is Codex Vaticanus, two of the most important manuscripts available.