1. Y-chromosomes are chromosomes, and can cross over when two are present. There are men that have three sex chromosome, XYY. In most of these cases, the Y chromosomes are duplicates, thus of the same haplogroup. However, in some cases that is not true. An obvious example would be double fertilization (when an egg is penetrated by two sperm cells, each with abnormal numbers of chromosomes). If the two sperm are from two different men (of vastly different Y-dna haps), the Y-chromosome crisscross would result in chromosomes that do not fit standard haplogroups. Also, this can occur if the female has a Y-chromosome (which can happen, if the Y-chromosome has a SRY deactivating mutation).
Why would an ultra-rare situation like Y-chromosome combination negate the viability of YDNA as a tool for analyzing population movements? If it was a problem, then surely we could find examples of lineages following a combination event? I don't know of a single one.
2. Y-chromosomes may be quickly mutating, but a population's Y-haplogroup composition is quickly changing. This is why Y-Adam lived much earlier that Mitochondrial Eve.
Y-Adam is currently estimated at 208,300 years old, and that's
a significantly reduced revision. What's the current estimate for Mitochondrial Eve? I recall about 200,000 years old, which would put them in the same ballpark.
That said, it is a good rule of thumb that Y-DNA lineages are more likely to go extinct or bottleneck than mtDNA lineages.
When two populations join, their MtDNA hap ratios will remain quite stable, while one Y-DNA hap will usually prove to be more dominant that the others. That is why many of the Y-haps belonging to famous ancient populations were not preserved. For example, when the Proto-Indo-Europeans and the Iranian U2e joined, the U2e's own men went extinct. This phenomenon is due to the potentially high reproductive ability of men. While women can only have one child within a certain time period, many men of a certain hap can have their offspring occupy the wombs of most of the women, thus denying the other hap a chance for reproduction. Also, the boundaries of Y-DNA haplogroups are much more blurred, since women tended to stay in one area, or move along with the tribe, while men were more likely to go on lone voyages.
Much of what you say about Y-DNA is true, but I think it's possible to account for it simply by knowing ahead of time that Y-DNA distributions tend to magnify the effect of migrations. There is a pretty important benefit to the frequent Y tree pruning, though, and that is that individuals lineages become more easily associated with particular migrations. Suppose we want to determine if a British lineage is of Anglo-Saxon or Celtic descent. We often don't have to go very far down the Y tree to get a good guess, and we can often use nothing but STRs, which we can't do with mtDNA. On the other hand, since mtDNA distribution was relatively inflexible in Anglo-Saxon and Celtic populations, the two populations don't look all that different to begin with in their mtDNA, so to figure out the source of a lineage, we'd need to test it to a more specific subclade than exists in most databases.
If your point is more about figuring out the composition of a population rather than the origin of a certain lineage, it's important to keep in mind that mtDNA distributions have the opposite problem as Y-DNA, namely, that they tend to minimize the effect of migration. Autosomal DNA profiles are likely to lie in-between the two in most cases.