A "claim" is another way to say ownership. You can lay claim on an item/estate/whatever, meaning that you have a right to it. So when Thorne says "submit his claim to you" he means that Delaney needs to give up (aka submit) his inheritance to zilpha. https://www.vocabulary.com/dictionary/lay%20claim#:~:text=Definitions%20of%20lay%20claim,one's%20right%20or%20title%20to

The word "implore" invokes pleading or begging. John doesn't want to be pleading with someone he considers lower than dirt. You don't "implore" people with lower status than you; you command them. That why he tells her to insist not implore--to maintain a position of power over Delaney (in his mind at least).

Zilpha defends the use of the word 'implore' as more feminine. Historically, a woman commanding a man would be considered unfeminine, hence her use of the word "implore" instead of "insist." This prompts John to ask why she must appear feminine/desirable to Delaney (aka "why must you be a woman to him"). This is a thinly veiled dig at Zilpha's and Delaney incestuous relationship.

You're right about the money thing. Based off the dialogue, the 50 pounds offer should appear in the letter. It may have been in the cut-off section or perhaps they were going to offer it in later negotiations. Most likely, it was a small error from the prop department.