Is it a stretch to believe that Rose and Scorpius will become romantically involved? If not, would that at least be capable of being friends?
For the former part of the question: likely, but not very. As said before, a lot of our discussion will be based on a particular writer’s story, the plot etc, and the requirement to mould the characters accordingly.
In a Romance fiction, this idea is plausible, if you’re a believer of the school of “opposite attracts”. I say this, putting my trust in a Scorpius who is a most likely Slytherin, a bit liberal in his views seen in comparison with his grandfather and father, but a shares a something little in interests or skills with Scorpius. Also, why does this romantic involvement have to only take place within the walls of the castle?
It is equally possible that if Rose and Scorpius ever became involved with each other, this will happen out of Hogwarts. The reason for this is simple. Ron shows us in the Epilogue that he still nurses something against the Slytherin house, despite the fact that his tone was quite mild about the whole exchange. With a whole lot of the family being in Gryffindor, it is likely Rose date guys not in Slytherin. Here I’d quote something cmwinters said in Scorpius’ Character thread: “…and in fact that's what that Epilogue pretty much said about the next generation; that nothing really changes.”
True. Very true. So all those stories featuring Rose/Scorpius must base their romantic feelings on strong reasoning and carefully develop those to achieve convincing characterisation.
Moving on. Although there is no evidence of this, but as a girl I believe Rose would generally take on the views and manner of thinking after her mother. This puts her as a Gryffindor quite not like her Ron or Harry. She will I believe support her own house always, but not abuse the students other than in her own.
Therefore, to answer the second question, she doesn’t have to be the extreme of feelings, being romantically involved or capable of a friendship with Scorpius, but something along the lines of a civil acquaintance would do.
It is most likely that these students will study about the Second War in their History of Magic classes. Hoping that the writer of those books in not a prejudiced wart, under the guidance of a good professor, that class will finally do some good, you know. Open the students’ mind, for instance.
Does Rose Weasley seem like the type to pick up Quidditch (does she take after her father's love or her mother's indifference)? What position would she play?
More than enough Quidditch blood in the family, I guess. So, reasonable to think that Rose will also play. Bring in the fact that Rose has a three-year or more difference with Hugo, so Ron will take considerable interest in teaching his daughter the game, before he gets to teach it to his son.
But will she play on the Gryffindor Quidditch team? We can only anticipate.
Both will and won’t are equally possible.
Here’s the family history of Quidditch players:
Ginny: seeker, chaser
For Rose, the position of a Seeker or Chaser is most likely.
What kind of relationship do you believe she has with her younger brother? With her cousins?
Rose seemed like a nice, obedient and easy-going person in the Epilogue. She will be a good influence on Hugo, I think. If anything, they most probably will mirror the Weasley sibling relationship at their best. Which pair of sibling doesn’t have their little arguments. As a matter of fact, bickering or random bits of teasing would make their relationship convincing.
I see her as sharing a particularly close bond with Albus. An overall friendly relationship with her other cousins as well.
Do believe Rose will share her mother's anxieties and rabid behavior when in comes to school work, or do you see her as being more confident in her own abilities as a student?
I see her as being more confident. One of the reason, that Hermione was very anxious when it came to her homework (especially in the first three years) was her being a Muggleborn. She came to a strange world and wanted to be academically sufficient everytime. This made her study more than was always necessary.
Rose would not face the same situation. This is the reason, she’s more confident about her skills and intellect. Combined with Hermione’s guidance, I see her as being responsible about her studies but never freaking out.