If you've received a manuscript/synopsis request from an agent you've queried (but haven't heard back from them about it yet), is it worth mentioning in the query letter to other agents? Maybe to bump their interest that someone else is interested? does it even matter?
No, typically you only tell agents when you’ve received an offer (and that’s as a follow up email to all the agents who already have your query/manuscript, not in the query itself). Getting an offer suggests that the entire manuscript was good enough for an agent to want to take it on, and that if they don’t look at it right now they might never have a chance. All a manuscript or synopsis request tells an agent is that your query and sample pages were decent enough. They also have your query and sample pages in their inbox and can decide that for themselves.
What if the agent I'm querying only wants a letter, and not sample pages? would mentioning the other stuff (that another agent requested more) be beneficial to make them curious enough to ask for sample pages, too?
Total novice here. My understanding is that unless you've received an offer, it's best to follow the agents guidelines. They all may be different, but you should avoid doing anything that questions their process, however well the intent is. Without working with any agents, I get the feeling they get so many manuscripts from aspiring authors they are just looking for any reason to reject a manuscript or business relationship.
All the agents I've queried have asked for a sample of the work, typically the first three chapters. Perhaps the agent you're querying deals with non-fiction submissions also and you're on that page of their website. The process of submitting a query for a non-fiction work doesn't normally require a sample.
Agents will ask for what they want, don't send them anything that they don't specifically request. It's going to turn a lot of them off. They all have their own requirements, make sure you follow them to the letter. You can't "entice" anyone. You can only send your best work and they will either be interested or they won't.