I've been up the Tissington trail several times on my road bike with 25mm tyres. Its okay if a little rutted in places (the main cutting in the lower half of the ride and then after the signal box). I do think a good pad is required however.
As to dog emptiers, they tend to stay within a mile or two of the "stations" so once you are beyond the one by the huge curve on the embankment (I forget what it is called), you will be on your own for a few miles. You will however, come across people who have hired bikes and aren't used to riding so its best to give them plenty of warning when passing!
As to speed, I won't claim to be the fastest person on this site but from the main car park just outside Ashbourne to the very top takes me on average 90 minutes for the 17 or so miles. There are a couple of sections of climbing (long false flats really) but the real test comes from the wind....once on the great bend, the wind picks up and can blow you sideways! I suppose it could be good practice for the Fens
With regards to going down the trail...that's the best bit
(My best time of 57 minutes was done at 7am in the morning when I had the trail to myself. I would not go that fast normally).
Finally, I agree about the A6 out of Buxton. Later in the evening may be okay but otherwise it's best to take care. Also, don't miss the entrance to the trail as there is a nasty hill soon afterwards!