How to continually email references without annoying them?
People who have agreed to be your referees know you will be applying to various things (or should do: you should have given them a reasonable outline of your plans). The number of different things they need to do is not so important. What matters is the total time it will take them. If they have three references to write at once, they can copy/paste it across, having got into 'writing your reference' mode just once. If you time requests such that they have to keep coming back to it, that will take much more time and effort for them.

So, do your best to group any reference requests into as few blocks as possible. That means you need to plan ahead about your applications, and ask for some well ahead of time, so they can be done along with references for places with earlier deadlines. As a bonus, this also massively reduces the 'night before the deadline' worry.

Personally I would scrub round the 'did you send it?' email. If it's after the deadline, there's not a lot you can do about it either way. If the place you are applying is very strict on the deadline (ie including references), make sure you have told your referees that. Unless you know they are particularly unreliable (in which case you probably want to take a different approach anyway), they will not mess things up for you except in an emergency. Some places will accept references slightly after the deadline, and the nicer ones would let you know that not all your references have arrived, so you can give the referees a poke. Also, if you've given them things in a neat block, there's a good chance they will let you know they've sent the references by themselves.