[R Gibert]

I'll respond to (1) only.

First off, if you are crediting him with a pocket pair, then there are 45 unseen cards rather than 47. The approximation for the probability you will catch one of your n outs with 45 unseen cards is:

P = 4.44*n - nC2/10 = 4.44*5 - 5C2/10 = 21.2%

To adjust for r redraws, you use the following simple approximation:

Q = n*r/10

This works for 46 unseen cards too. For 47 unseen cards, the divisor is 11.

However there is a complication. The number of redraws varies depending on whether you make trips or not. Your opponent can make a higher 2 pair by pairing one of the 2 flop cards you are not paired with. The calc for Q must be split in 2:

Q = 2*2/10 + 3*8/10 = 2.8%

The 1st half says that for 2 of your outs that make trips, your opponent has 2 redraws that make higher trips. The 2nd half says that for 3 of your outs that make 2 pair, your opponent has 2 + 3 + 3 = 8 redraws making trips or 2 pair.

With this adjustment we can now correct your winning chances to be:

P' = P - Q = 21.2 - 2.8 = 18.4

Using poker stove with a hand like 7s6s vs AsAh and a flop of Kh7c2d gives 18.384%.

If you had blindly used the rule of 4 and ignored the redraws e.g. with 5 outs 4*5 = 20%, you would have been close enough for most purposes. The smaller number of unseen cards partly offsets the effect of the redraws. However, with a larger number of outs and/or redraws the difference can be more significant.

Note that my calcs and example hand sidestep the further complications of backdoor draws.