A great deal of the Anglo-Saxon upper parts of society were wiped out in the battle of Hastings. King Harold, two of his brothers, and an unknown, but certainly sizable, amount of the English huscarls and other soldiers were killed in the battle. His remaining male relatives were either imprisoned by the Normans, in the case of his brother Wulfgar, or fled, as his sons seem to have, to Ireland and Wales. In the years after the battle of Hastings, several Anglo-Saxon lords rose up in rebellion, Morcar and Edwin, both northern Earls rose up in rebellion in the years after and were defeated, Edwin died during the rebellion, and Morcar was imprisoned, his ultimate fate however isn't really clear. Eadgar the Ætheling also survived the invasion and an abortive attempt to be crowned king, and ended up with quite an elaborate career or inciting the Scots to attack the Normans, serving in William's court, and according to Oderic Vitalis, participated in First Crusade.

While this sums up the nobles on the Anglo-Saxon side what about the others below them? While our sources are usually quite silent on the number and names of the individuals who were left leaderless in the wake of the invasion, sources do paint a rather fascinating picture for at least some of these men.

According to disparate sources, a great number of Englishmen during this time frame elected to leave England and seek a new life in the Byzantine Empire. While I am loathe to lean on literary sources such as sagas, this information is also recounted in other sources as well. Anna Komnena indicates that the Varangian Guard took on a decidedly English flavor in the years after the Battle of Hastings, where these men eventually found themselves fighting against Normans once again, as a part of William Guiscard's invasion of the Byzantine Empire. A great many of them perished in the fighting according to the Alexiad.

Two other sources claim that English expeditions reached Constantinople and were given land on the Black Sea by the Emperor Alexios, and founded a new settlement there. While the claim is certainly fanciful, the influx of Englishmen into the Varangian guard following the Norman conquest is not particularly controversial.

This covers a great deal of the men of means who were involved in the Norman invasion. Its likely that quite a few fell in the rebellions against William or fled England altogether. Many scions of the nobility and aristocracy ended up in Scotland, Ireland, or in the case of Eadgar, across Europe.

The other part of your question is a little trickier. Military equipment at this time would have been fairly similar across these cultural divides, and there would not have been significant differences between Norman and Anglo-Saxon military gear. A great deal of equipment would have been personalized, but the equipment of an Anglo-Saxon huscarl would not have been outrageously different from the equipment of a well to do Norman soldier either. The only notable difference in equipment would be the two handed axes that the English (and Scandinavians) were famous for wielding, but the swords, shields, and mail of the two groups would have been very similar. This is largely an argument made of speculation though, since very little mail or weaponry survives from this time period, and historians are largely working off of textual and artistic evidence, and not archaeology.

Military gear at this time though wasn't made or maintained by large state run armories. While the king and lords would have maintained a sizable contingent of soldiers bearing weapons and armor, this equipment wasn't standardized and able to be recycled among various groups. When an English retainer left for Constantinople, he almost certainly would have taken his weapons and armor with him. While some gear was undoubtedly scavenged from the battle fields, there is no way to extrapolate how much Anglo-Saxon equipment made its way into the hands of men serving and fighting under the Normans.

Now the last part of your question I cannot answer in full because I do not know a great deal about mercenary companies and groups in this time frame in general. Sources are generally silent on the nature of William's army and its composition following the battle of Hastings. What role Englishmen may have played in it is largely obscured at this point.

---

Sources and further reading:

I've used bits of a few primary sources, Oderic Vitalis's Ecclesiastical History (Don't let the name fool you, he's an excellent source for this time period), Anne Komnena's Alexiad, and there's a few others that i could point you towards if you have more specific questions.

For secondary sources,

Betty Hale, "History of Prince Edgar & his Claim to the English Throne"

Donald Henson, "The English Elite in 1066: Gone but not Forgotten"