Well, I've been busy this week, but I wanted to put down some notes on my focus before Week 1 comes to a close. I haven't read many of the comments so I apologise if I'm re-iterating what has already been stated.
For my focus, I've decided to look at how family and friends influence the characters' choice of a marriage partner.
Much has already been said about LR with Anne and CW, so I'll just add that Sir Walter opposed the match primarily due to CW's lower social status, whereas LR had issues with the finances and CW's personality. LR has more influence on Anne due to a closer relationship, and Anne believes that LR's reasons are well founded.
LR couldn't convince Anne to marry Charles Musgrove however due to his comfortable situation in life. The text doesn't seem to give any indications as to why she was not successful.
Sir Walter does not expect Anne to marry as she is too plain. He indirectly influences her in this by not taking her to London to put her in society to meet a partner.
The Kellynch community assumes that LR will want to marry Sir Walter, but neither feel a need to - LR because she is already respectable and provided for, and Sir Walter for Elizabeth's sake.
Sir Walter expects Elizabeth to marry well, and wants her to marry Mr Elliot. Elizabeth wants to marry him, but it appears most of this was due to being talked into liking him:
... Elizabeth, who had liked the man for himself, and still more for being her father's heir...
This comes after only meeting him a few times.
Sir Walter is personally offended when Mr Elliot marries according to his own wishes. The offense seems to be that Mr Elliot refused Sir Walter's influence:
As the head of the house, he felt that he ought to have been consulted.
Once again, this is absurd given the few number of times Sir Walter has met Mr E.
To wrap up, the cases where persuasion works is when the person being persuaded has a particular bond, and the advice given fits with their general world view.