1) Should I limit my warmup and cool down more?
2-3 miles is fine. Not warming up is a pretty dramatic reaction that's inviting injury. The kinds of injuries you see with this work can be severe if they happen, but are almost completely avoidable with a decent warm-up.
While I get the idea to limit it to be even more fresh, most of us are so accustomed to our routine that it adds no legitimate fatigue to your system.
2) Trainers, flats, or spikes? I have been wearing trainers for hill sprints and flats for the flat sprints but should I transition to spikes at some point?
If you're trying to run fast you need to set yourself up to run fast. Eventually, this kind of work should be in spikes and on a track. I don't see any downside to spikes here.
3) Measuring distances? This might be a stupid question but how would you measure out for example 60 m on the track. Is there a marking somewhere or just use a wheel/football markings?
the closest mark to 60m is probably the women's 100m hurdle mark at 61.5m from the finish. That's their second hurdle mark in the race. To be honest the distance doesn't really matter. Is it long enough to hit top speed? is it short enough that you aren't losing speed at the end? if you're in that ballpark you're good.
4) Once I’m into my races or time trials this fall (probably focused on 3k-5k) should I move these workouts to once every two weeks? And should I stop them when tapering?
I do them once a week pretty much year round and take them out during tapering (by that point the hay is in the barn). My exceptions are when another workout takes it's place (speed endurance or a really fast SE1 day) or when we're doing hills instead of flat sprints.
5) Do I need to incorporate true speed endurance workouts if focusing on 3k/5k this fall? Normally the shorter rep workouts that we do in the fall are 200-400 reps that are at mile pace or slower. How would I start to incorporate this if I should?
Everyone's got different takes on this but my kids even up to 10k do these sessions. They might be more spaced out and probably shorter or less particular about how fresh they are, but we've had some success with it and I really don't think it takes anything away from a plan just to tack on 4-5 fast reps after a mid week run. We know it's not true "speed development" to do it after an "easy 10" or as part of a long double day, but sometimes we have to go with what's good enough. 10k runners do have more important things to worry about.
Think about it like a vitamin. You might not need a ton of it and it may not have any immediate effects, but it's probably not awesome to have none of it.