Does it count as a set if you stand up after each repetition during deadlifts (to take a deeper breath and brace more properly) which also includes resetting your grip?
Background: I am training for powerlifting
It's a deadlift, meaning the weight should be dead on the floor. Touch-and-go or otherwise bouncing around is not a deadlift.
Personally, I relax my grip and reset on every rep. My hands don't come off the bar, but I open my fingers, ensuring that the weight is indeed dead on the floor.
The extra second this adds to each set is negligible and the benefit to ensuring proper form far outweighs any loss. You also need to breathe between reps, and the bottom relaxed position is the perfect spot to dump the air in your lungs, get a nice breath, breath that out, then get another, hold it in, tighten the abs, get your chest up, and pull.
If you are working with enough weight that you need multiple breaths to get each rep, take them, but don't just stand there and pray that the next rep will be easier. If you need 2 or 3 between, take them.
So for those 2-3 breaths between reps, you have plenty of time to relax your grip, brace for your next lift, and make sure you're lifting clean.
If you're deadlifting near your 5RM, your biggest concern should be proper form. If that takes you 1 second, 5 seconds, or 10 seconds, take the time.
Regarding standing up in between, that might be a bit excessive, but if it's what you require to nail your form, then do it. For me, I don't want to get my hands off the bar and I want to stay mentally in dead-lift mode. When I stand up without a bar in my hand, I'm done with my set.
The problem I could see with standing up is that you are probably moving your feet as well, so you'll be fully resetting between each rep. If that's a net positive for you, causing you to have dialed in form, good breathing, and hit your numbers, then by all means stand up. But if standing up is a bit too much of a "break" as it is for me, then it might not be serving you well.
John's answer is correct - all in all, it's hypertrophy (TUT) vs strength (rest).
I just wanna add that it's a matter of how long the stoppage lasts, and not the reason that made you stop (as long as you're remaining in the starting position). e.g. resetting your grip is equal to stopping for a breath.
There is no definition for how long the bar should lay on the floor between reps, but I'd say that the process of stabilizing yourself in the starting position and start getting up shouldn't take more than 2.5 seconds. Here's a very random example I found that represents a good resting time between sets (1.5-2 seconds). When you go really heavy, for a set of 2 or 3 reps, the resting time goes up to 4-5 seconds, as I saw in some pro's videos.
One last thing to mention, in competitions they perform 1 rep at a time, so for this subject your training habits don't matter.
Yes, depending on your goals
Resetting your grip likely takes a bit of time, perhaps a few seconds, perhaps 10 or more. During this time, your hamstrings are (mostly) resting. This changes the results of the exercise somewhat.
From a hypertrophy standpoint, the more you are 'resting' between reps, the more you're losing a bit of Time-Under-Tension, which means the muscles will have a little bit less stimulus to grow.
From a strength standpoint, the more you rest the more muscle-fibers will be ready-to-fire, so you'll have more strength to pull with.
In other words, if you're taking enough time to reset your grip so that your 5x5 protocol is becoming more of a 1x25 protocol, it's going to change what you get out of your workout.