Caveat lector: I do not have Shakespeare's phone number, nor does he come to me in visions, and you do not have to believe my nutty ideas. But you might want to consider whether they make sense in the context of the play. Enough timorous prevaricating! Here goes:
After I had been familiar with the play for seven years, a private student, Bonnie Halligan Ryerson, commented on the Countess' line, "Even so it was with me when I was young." Bonnie said, "Gee, maybe the Countess had an experience like Helen's." A light bulb went on over my head as it does for characters in cartoons. Bonnie snapped a picture of the event.
When the Countess and the King and Lafew were all Bertram's and Helen's age, the (unmarried) Countess and the (unmarried) King fell in love. Their fathers absolutely forbade them to marry (royalty can't marry anybody but royalty, unless her name is Middleton). So the King and Countess made plans to elope. Probably unwittingly, Lafew, the King's good friend, let slip the plan to their fathers. The fathers flew into rages, and the Countess' father immediately married her off to Count Rossillion. Neither the King nor the Countess had the courage to defy his/her dad. As a result, both King and Countess wound up married to people they didn't love.
"Is that why Lafew always kneels and begs pardon when he comes into the King's presence?" It is indeed. I doubt that Lafew wrecked their plans on purpose, but he did do it, and nobody has gotten over it. For the King, forgiving Lafew has become a pro forma affair; but Lafew never dares to forego his apology.
"Why didn't they just cheat on Count Rossillion?" Because they are both honorable people; because he was the King's good friend, a good and truly noble man, and neither of them had the desire to betray him. So they have lived for 18 years or so since then, still in love with each other, unable to do anything about it. No wonder the King says to Bertram, "My son's no dearer" [than Bertram; I.2.76].
The King and the Countess see in Helen and Bertram a chance to fix the past. They want Helen to marry Bertram. The Countess is so tough on Helen (in I.3) because she wants to make sure that Helen will have the courage to go through with her plan to marry Bertram, despite any obstacles (the Countess gave in before her father's wrath, and agreed to marry Count Rossillion, and has always regretted her weakness). When Bertram says he won't marry Helen because she's a commoner (II.3.113-6), the King flies into a rage like that of God in the Old Testament, and forces Bertram to submit. King and Countess see their young selves in Bertram and Helen, and this time they are determined that the marriage shall take place. And in the last scene, Shakespeare has put both King and Countess on stage, both regretting their ruined pasts ("That's good that's gone" V.3.55-70).
It's also the reason the Countess is so rough on Bertram (as in her less-than-full-throated blessing, I.1.57-66): he is not the son of the man she loved, and still loves.
I don't think you can make this story clear to the audience, but it should be a clue to the director and actors why the Countess rages furiously at Helen ("Only sin and hellish obstinacy tie thy tongue" I.3.174-5; you can see our video of the scene at https://www.youtube.com/watch?v=QiMrN_1uuJY&feature=youtu.be) and why the King fulminates at Bertram like Jehovah ("Or I will throw thee from my care FOREVER!" II.3.162).
Incidentally, everyone on stage knows that the King and the Countess have a history (even though we don't), and the knowledge affects their behaviors. When Helen tells the king that she wants him to give her a husband, the King may or may not look frightened, but in any case Helen is quick to reassure him that she doesn't want to marry the Dauphin ("Exempted be from me the arrogance / To choose from forth the royal blood of France," II.1.194-5).
If you produce the play, you may not skip over Lafew's kneeling, but it will remain a mystery to the audience, like Helen's "etc." But I always think a play is more interesting if the audience doesn't immediately understand everything that's happening on stage. It pricks up their ears and rivets their attention.
Back to the Introduction.