Colonel Brandon has his own room at Barton Park, so he is quite accustomed to spending time there, even before the Dashwoods came. More than once he is called the 'particular' friend of Sir John, and it is clear that he has spent a lot of time at Barton Park over the years. Their friendship is a lot closer than just 'polite' and so it is not remarkable that he would spend four days at the Park in the course of two weeks.

Actually, there is proof in the novel that he did not fall in love with Marianne at first sight. From Ch. 10:

Colonel Brandon's partiality for Marianne, which had so early been discovered by his friends, now first became perceptible to Elinor, when it ceased to be noticed by them. Their attention and wit were drawn off to his more fortunate rival; and the raillery which the other had incurred before any partiality arose , was removed when his feelings began really to call for the ridicule so justly annexed to sensibility.

This is clear. When they first started teasing him about Marianne, he did not yet have any paritality for her. By the time his feelings began to be what Sir John and Mrs. Jennings had 'accused' him of, Willoughby was already there and they had dropped the teasing towards him.

He doesn't fall for her because she is 'taken'. He falls in love with her because, as he gets to know her, he realizes how much her sensibilities are like his own. His regard for her continues to develop throughout the story.

People in that time did not date. 'Courting' someone was with the intent of marriage. If Brandon, on a few days' acquaintance, started 'courting' Marianne without even being sure that he was in love with her or wanted to marry her, that doesn't make him confident so much as a number of other very undesirable adjectives. He's been burned badly in love before. He's been set up with a number of other young women with Mrs. Jennings as matchmaker. I can't imagine that he would jump into a relationship he wasn't even sure he wanted on four days' acquaintance.

If there is no evidence in the novel that Eliza named her child after herself, then there is also no 'evidence' that anyone in the novel is using the names their mothers gave them, either. In that case, maybe they are all running around under assumed names. It begins to get into rather extreme conjecture.

We know the little girl was called Eliza, we know that Colonel Brandon never met her or saw her until she was 3 years old and we know that people are named by their parents. Children are often named after their parents: John Middleton, Thomas Palmer, and several of the eldest daughters named after their mothers in the other novels.

In absence of any shred of proof to the contrary, it is a far more reasonable assumption to make that she was named that by her mother. If Jane Austen intended to make the point that Colonel Brandon had changed her name and called her after her dead mother, then she certainly would have said so.

'Timid' does not seem to be a character trait one could attribute to the type of person who duels and tries to elope.