More specifically, how did his forces compare to those of Harold Godwinson and William the conqueror? Did William have knowledge of Hardrada's invasion of England before the battle of Stamford bridge, and did he have plans to fight Hardrada in the case of a Norwegian victory? Were there any other claimants to the throne? I hope this isn't against the rules, but what do you believe were Harald Hardrada's chances of success in the war? and of course any other information that you think is relevant to Harald Hardrada/Stamford bridge would be welcome.
Hadrada most likely had a roughly equal size force compared to Harold, circa 10,000 men. As to whether William knew or not, he may well have had men in England who could inform him of such an invasion, hence his convenient arrival just as Hadrada was being routed.
William almost certainly anticipated fighting the Norwegians should they have won, which was a possibility, not only did he have a similar force, but Hadrada also was being aided by Tostig, the ex earl of Northumbria and Harold's brother seeking revenge for being deposed of his title. So I would say a victory was indeed plausible. As to other claimants, Edgar aetheling - Edward the confessor's nephew - also had a legitimate claim, yet was only 14 and thus had lacked the support of the witangamot (wise council, yes this is like the wizengamot from Harry potter) and reportedly was sickly and ill disposed to rule.
The main cause of the problem (IMO) was Edward the confessors lack of children, there is no one answer why he didnt. He may have been Gay or perhaps infertile, had taken a vow of chastity