She didn't force any of the animals to participate.
They just wanted to be her pet that badly.
It was somewhat disappointing how she goaded them on, though. Then again, that's kinda what being Rainbow Dash's friend is like. Somewhere beneath all the prickly is a gooey marshmallow nougat. You just have to put up with her being a ***** all the time.
Incidentally, Rainbow Dash does slightly change her attitude around each of the ponies. To wit:
Applejack is super tough, and not afraid to tussle if need be. Rainbow Dash can play as rough with AJ as she wants, and she'll get the same roughness in return. This is consensual in the case of both parties, though I think their friends don't fully understand the nature of the relationship. I don't think it's entirely unhealthy either, although traditionally speaking this is a very masculine type of friendship, where physical conflict is usually not a sign of the end of the friendship. But then there we go with FiM vaporizing yet another gender stereotype.
Twilight gets a lot of verbal abuse from Dash, but Twilight's shown a certain resilience, or maybe indifference, to Dash's name-calling. Usually Twilight just blows Dash off when the latter gets choppy. I don't know if this is an unhealthy streak in their relationship or just a verbal equivalent to Dash and Ajay's more physical sparring. I do know that Dash's verbal abuse veils a certain respect for Twilight's leadership skills. It's possible Dash doesn't really know how to verbalize that respect. And Twilight puts up with Rainbow Dash because she knows she can rely on Dash.
As for Fluttershy, it's worth noting that while Rainbow Dash plays rough with Twilight and Ajay, she never makes fun of Fluttershy. Anything negative she says is an earnest criticism. The relationship between Fluttershy and Rainbow Dash is so interesting in this way. I think Fluttershy is one of Rainbow Dash's closest friends, and yet Rainbow Dash, being who she is, and having dealt with Fluttershy's social paralysis as long as she's known her, is probably just immensely frustrated that her friend continues to hobble herself so. In the end, Dash's blunt criticisms of Fluttershy come from a deeper urge to see her friend gain confidence than from mean-spiritedness.
Dash is also not particularly mean to Pinkie Pie. They exchange pranks and the like, but Dash seems content to laugh with Pinkie Pie more than at her. I think Dash just realizes it's almost impossible to make fun of Pinkie Pie for anything, because if there's a joke to be made about Pinkie Pie, Pinkie Pie's already made it.
I'm not yet convinced Rainbow Dash knows Rarity exists. This amuses me because Rarity is all about some Rainbow Dash. Joking aside, I think Rarity is just an enigma to Dash, and Dash doesn't like enigmas, so she just lets Rarity do her thing, and tries her very best not to get involved. I really, really want to see an episode where these two are forced to interact, because there's pretty much zero excuse for incidental interaction between the two. From what little we've seen, there's potential for tension, but very little actual animosity, unlike the relationship between Applejack and Rarity.
Basically, I think Rainbow Dash was looking for a pet with whom she could have a relationship like her one with Applejack, or more illustratively, Gilda, pre-breakup. This is why she was so abusive. She wanted a pet that could "keep up" as far as playing rough. Tank wound up forming a totally unique relationship with Rainbow Dash. He and Dash both know perfectly well what an odd couple they make, but they're both tough enough to put up with the strain it causes because they care about each other, or something mushy like that. I actually have to applaud the writers for going somewhere totally new with the concept.