How do I manage this without being disrespectful?
Just ask. Ask at least a month before your first deadline. (Ideally, your writers should be people who know you well enough not to be surprised by your request.) If possible, ask in person. Give them as much information as you can about where you're applying, including your complete application package — CV, statements, sample papers, names of your other references, everything. Ask for feedback. Listen. If they agree to write you a letter, keep them regularly updated as your CV and application targets change.
Most importantly: Given your references a way to tell you which letters they've sent. This could be as simple as "Please send me email when...", or a private Google doc listing your application targets and deadlines. (From personal experience: Having a common Google doc for all your letter writers is particularly effective, since each writer can see when the other writers have sent their letters.) Check regularly that your references are actually sending their letters, and bug them mercilessly if they haven't yet. This is not being disrespectful; this is helping them to do the job they agreed to.
Finally, keep your references updated whenever you get an interview, get a job offer, or accept a position. Don't forget to say thanks.
How many requests in a given unit of time is too many, or harms my credibility?
That's entirely up the the individual writer. But as Rex says, it's common for one applicant to apply for dozens of positions. When I agree to be a reference for someone's faculty search, I generally expect to send letters to 30–50 different targets. I can use (nearly-)identical letters for most targets, but inevitably there are major variations. For example, tenure-track faculty positions, industrial research lab positions, and postdoc positions all require different letters, even when the applicant is applying to all three types of jobs.
Moreover, there is no standard mechanism for submitting letters. Some places want letters submitted in parallel with the application; others ask automatically when the application is submitted; others ask only when the application passes an initial filtering stage. Some places want letters by email; others use standard services like MathJobs; others use home-grown web pages; a rare few still require paper letters with wet signatures.
This is what agreeing to write a job letter means. Most faculty already know this and won't agree to act as your reference unless they're willing to do this much work. Still, just as it's important to ask "Can you write me a strong letter of recommendation?", it's important to be completely up-front about the scale of your request.