From everything I've seen, it's all anecdotal evidence. I certainly haven't seen a decent study of Doctors vs team strength vs injuries. I've looked into hundreds of games in various aspects to get a better handle on game mechanics, but I personally haven't actually recorded injuries and known statistics to find any solid correlations. It would be an interesting study in any case.
From what I've experienced and seen in various games I've looked over:
1 - My personal luck in Scrimmages are horrible. Almost all my injuries come from them. However, looking at multiple other teams and their scrimmages, it just looks like bad luck for me and that no particular weight is given to these games. Cup games are even harder to tell as the sample size is so small compared to normal season games.
2 - From what research I've done into it, it seems to occur more when there is an offset in player strengths as opposed to Strong vs Strong. From what I've seen in various games it appears that weaker teams tend to rack up injuries very easily. This could also simply be a factor of the stronger team tending to have a higher Doctor level so they don't experience the injuries that the weaker team pulls down. Part of the problem with studying this particular issue is that there are so many unknowns. No way to know any Doctor level but your own. No way to know player aggressiveness. No way to know Enthusiasm for any given team. Any or all of these could have a definite impact on injuries.
3 - I haven't noticed this. From what I've seen, injuries tend to occur more often on larger mismatches, but not necessarily to a weaker player. IE, a 10K Center vs a 140K Center is a bit of a danger, but a 5K PG vs a 5K PG doesn't seem to have the same effect, even though the 10K Center and 5K PG are on the same team and playing at the same time.