A) When you say it’s for a court case and looking for expert testimony… why are you using the internet? You should be talking to a scientist or doctor in person. In the end, if it’s a matter of “who’s the daddy?” it doesn’t matter how it happened. Just do a DNA test on the kid.
B) Sperm are cells. They need a certain temperature to live in, food, and moisture (among some other things). I would say that the longer they are outside the body on that shower floor, the less viable they would be. Once they leave the penis, they have to swim like a bitch to get to the ovary (if they aren’t destroyed by the vaginal canal). The longer they live in an unfavorable environment, the more likely that swimming ability is going to be destroyed, if not the other important aspects of the cell itself.
C) It’s a simple matter of pressure in terms of “does water go up into the vagina.” Water may get into the vaginal canal but it’s not going all the way to the uterus. The cervix is generally tightly closed and if nothing else, think of putting a cup upside down into a tub of water. The air bubble inside keeps water out. If a girl is masturbating in the tub, water might get up into her canal, but it’s certainly not going up into her uterus. That would probably be dangerous anyway.
D) The semen on the finger is a legit way of getting pregnant. It’s much more likely to happen than this bathtub thing. It has lower chances than full intercourse, but it’s still doable. I put it on the same probability as the “pull out” method of getting you pregnant.
E) While it does technically only take 1 sperm to make a baby, it requires a hell of a lot more to get someone pregnant. Getting through the acidic vaginal canal, requires the death of a bunch of sperm to let the others through. Getting pregnant after exposing your vaginal to 1 sperm cell is near impossible. I won’t say it is impossible because we haven’t tried it, but it’s pretty damn near impossible.
F) Women do have a “flow”. The uterus contracts to help the sperm get up into the Fallopian tubes.
G) I have not actually worked in a fertility clinic, but in a science laboratory, if you store the cells in a medium with the proper nutrients and a few other chemicals (including anti-freeze), you can freeze cells and then thaw them later. They will live on as normal cells after that. I would assume the fertility clinic uses something similar to that.
In the end, possible but super incredibly unlikely. The sperm being outside of the vaginal canal to start really really decreases the chances. They would be diluted, they would need to know which way to go (and there are chemical trackers to help that normally, but they are likewise diluted in a tub masturbation), and they’d need to survive long enough to somehow make it to the uterus.
What @Parrappa said.