Learn English – what does “provide for” mean

In this connotation, it means "in fashion" or "in the know", and is sometimes used similarly to the term "cool".

Advanced Learner's has a good definition:

knowing a lot about what the most modern fashions are, esp. in music, social behavior, and styles of clothes

At least in my experience, "I'm a sucker for X" means that I am drawn to X regardless of what other characteristics X may have. The connection to sucker meaning something like loser, therefore, is that someone who is a sucker for something may get into a bad situation as a result, or at the very least enjoys X to a degree that seems injudicious and excessive.

This source, which does not seem like the greatest source, nonetheless offers three usage examples which seem fairly typical to me:

1. I'm a sucker for a pretty face.

2. Ted is a sucker for any dessert with whipped cream on it.

3. I don't know why I volunteered for this job. I'm a sucker for punishment I guess.

All of these illustrate the concept that to be a sucker for something means to be attracted to it while being somewhat blind to its other qualities, suffering as a result. 1) would be most likely uttered in a context where the speaker is justifying how he got involved in a regrettable relationship - he shouldn't have been with this person for some reason, but he's a sucker for a pretty face. 2) implies that Ted not only likes whipped cream, but will automatically eat any dessert with whipped cream on it, no matter how tasteless the amount of whipped cream. 3) has the speaker sarcastically suggesting that, since her job is so unpleasant, nobody would have volunteered for it unless they were a sucker for punishment.

To answer your question about formality: this construction is so common that I would hesitate to call it slang, but it certainly feels a little informal. In your specific example scenario - speaking with a client - I would not use it.