This is an interesting topic. A few responses...
Seems they had to "make do" right from the start, as William was illegitimate himself, so wouldn't have had a claim under Salic law either to Normandy...
Salic law wouldn't have worked several times over in British history.
William the Conqueror was an invader, but his male line died out very quickly - he had 3 sons and 2 male-line grandsons, and neither of his grandsons outlived William's youngest son, Henry. Within a generation William's male-line would have died out, creating a crisis (even without Salic law it created a crisis). Salic law wouldn't have stopped William from coming to the throne as he was a conqueror, but it would have prevented his descendants from maintaining it.
Both Stephen and Henry II got their claims through female lines. Similarly, Edward IV's claim came through female lines - with Salic law there would have been no War of the Roses, as the Yorks wouldn't have had the same leg to stand on. Henry VII would have also been disqualified, as his claim came through his mother. So would have Mary I and Elizabeth II, due to their gender, and James VI and I, as his claim also came through his mother.
Mary II and Anne would have been out due to their gender, and William III's claim would have been out because it came through his mother. So would George I's claim, as it was through his mother and grandmother. Victoria couldn't have reigned, nor could QEII, and Edward VII would have been out because his claim came through his mother. There is literally no way of knowing who would be monarch now because female descent has come into play so frequently.
IIRC, Henry was slated to enter the church, and would likely have risen to the highest ecclesiatical powers; most probably, he would have served on his brother's Privy Council, and perhaps been named Lord Protector in the event of his brother's death during the minority of Arthur's children.
He'd have made a horrible priest, though; he did seem to, ahem, like the ladies. He'd probably have fathered numerous illegitimate children which, if Arthur's children died or if Arthur left only daughters - or, if Henry were such a protector as was Richard III - could have resulted in a War of The Red and White Roses.
Was Henry slated for the church? He was only the second son, and his immediate ancestors don't seem to have slated many sons for the church as it was. If he had gone to the church he would have done fine as a clergyman - his womanizing wouldn't have been an issue because regardless of celibacy vows, many priests had mistresses, and Henry was extremely passionate about his religion.
Had Arthur fathered a son before dying I kind of doubt Henry would have usurped the throne. It's possible, yes, but he knew his history and I think he would have realized that England would no longer tolerate such behaviour - it might have been one thing in the days of Henry I or Richard I and John, but another come Henry VIII's time. A big part of what caused the problems in Henry IV and Richard III's reigns was that they were seen as usurpers. To usurp the throne would have caused Henry VIII huge grief, when as Lord Protector he would have already been very powerful.
Had the House of Hanover established the Salic law, which was used in Hanover istelf, the UK's present monarch would've been a 25-year-old named King Ernest Augustus VI. Among the other members of the royal family would be HRH The Duke of Windsor (if we assume that the Prince of Hanover would've gotten this "special" title upon marrying a Catholic and thus abdicating the throne), HRH The Duchess of Windsor (Caroline of Monaco), HM Queen Monika (step-grandmother of the "king"), and HRH The Duke of York (Prince Christian of Hanover). Pretty interesting, isn't it?
The Hanovers would have had a hard time justifying introducing Salic law to the UK given as their claim to the throne came through two women - Sophia of Hanover and Elizabeth of Scotland - in the first place, not to mention the fact that Elizabeth I was widely recognized as a good monarch. By the time of the Hanovers, too much history had happened for Salic law to be introduced into Britain.
If Mary I's pregnancy had resulted in a child, and the child was a boy, England and Spain would be more closely tied than they are now. However, if Elizabeth I had married the Count of Anjou and had children, James I/VI would have simply remained King of Scotland and Elizabeth's surviving descendants would be on the throne. If Edward VIII had married a suitable woman and fathered a legitimate heir, Elizabeth II would never have been crowned Queen. There would probably be a King Edward IX on the throne instead...with perhaps another George, Prince of Wales and then the whole Charles and Diana thing probably would've never happenend...because...well...Charles would have been George instead.
If the salic law had been changed in Hanover, Queen Victoria would have ruled over Hanover and the UK, thus QEII would be ruling over more territory.
It's speculated that both Elizabeth I and Edward VIII were infertile, so even had they made marriages (or in Edward's case a more acceptable marriage) it's likely they wouldn't have had children, and QEII would still be Queen.
On the other hand, if William IV had married younger instead of taking a mistress, or been allowed to marry his mistress, things may have been different. He fathered 10 children by his mistress, and is the only English/British monarch who has living descendants and isn't an ancestor to Prince William.
Henry wasn't deranged at the beginning of his reign. His paranoia developed after he contracted syphilis, and he acted tyrannical only after his major jousting accident so I don't think history would have repeated itself.
I think the syphilis is just one theory as to the cause of Henry's paranoia - there are a few others. Mental instability seems to be something of a familial trait.