1. Is there anything in canon that says what lightsaber form Kylo uses?
2. Why does he swing his crossguard saber like it’s heavy? Is this just the technique he needs to use to avoid cutting off his wrist with the crossguard? His form visibly changes in TROS when he switches to the Skywalker saber, so I assume that’s why.
Canon has been pretty stingy about lightsaber forms compared to Legends. To my knowledge, there is no official confirmation about the form that Kylo uses.
There is a ton of speculation about what form he might use however, and the general consensus seems to be a personalized variation of Form V (much like Luke and Vader/Anakin before him). A lot of his choreography is partially inspired by moves used by Luke, Anakin, and Vader. For example, the repeated overhead strikes call back to both Anakin and Luke, and the disarm he employs is borrowed from Vader's choreography in ESB. The reverse grip Kylo is often shown using is also commonly associated with Form V.
On top of that, he's displayed aspects of forms IV, VI, and VII. Episode IX especially displayed his use of acrobatics in combat, both during his duel on the Death Star remains against Rey, and during his fight against the Knights of Ren. People speculate he incorporates form VI because of his frequent integration of Force powers into his lightsaber combat, and because he's been depicted dual-wielding twice (once in TLJ, once against Ren in The Rise of Kylo Ren). The speculation of his use of form VII comes mainly from the unrestrained, raw aggression he displays while fighting, but I don't buy into it, personally. He doesn't seem to fight any more aggressive than, say, Anakin in RotS.
Lastly, regarding the change in his fighting style after switching lightsabers, I do agree that it almost certainly had to do with the crossguard, and possibly also in part due to the unstable nature of his red lightsaber. We know lightsabers "feel" heavy (despite the blades not having any weight). This is remarked on by Sabine in Rebels, and is sort of a carry over from some Legends sources. Something to do with the living nature of the blade, and the gyroscopic effects the containment field causes. It's possible that, since the containment field of Kylo Ren's red lightsaber seems unstable, it radically changed the way the weapon handles. That's just my personal speculation, however.
Oh, it's also worth mentioning that, iirc, some of Kylo Ren's choreography was based off of real world broadsword and longsword techniques. This was done to differentiate his fighting style with the crossguard saber from how other characters fought in Star Wars (typical lightsaber choreography was more heavily inspired by fencing, bushido, tennis [yes, really], and dancing). The idea seemed to be that the typical lightsaber was like a katana, while Kylo Ren's was more like a European sword.
Yeah, I'd say Kylo used a unique hybrid Form with no distinct classification. As UA mentioned, you can pick out several different Forms during his fights in the films: from Form I, to Form III, to Form IV, to Form V, etc.
When Kylo started using the crossguard saber, he naturally would've had to adjust his overall style a bit in order to compensate for a different/bulkier weapon(which makes sense, given that he essentially transitioned from a katana to a broadsword), but the root styles/Forms still seem to be there.
And tbh, Kylo's hybrid style probably goes back to his training under Luke, who had working knowledge of all the common Forms, and essentially invented his own style by incorporating elements from multiple Forms as well.
Apropos of nothing, but can I just say I love Luke's hybrid form? It's effectively the most balanced fighting style in Star Wars. The raw, simple efficiency against multiple opponents from Form I, the near-impenetrable defense of Form III, the battlefield mobility and hand-to-hand elements of Form IV, and the overpowering strength and off-balancing counterattacks of Form V. Sure, there are characters who know how to use all seven forms, but they typically are depicted transitioning from one form to another. For Luke, though, he employs all those elements together, with no need to transition between forms.
That's not me saying Luke is the best duelist or anything. Other characters certainly outshine him in different areas. And I know other characters, like Vader, have also blended many of the forms together. But to me, Luke just seems like the most well-rounded combatant. The only times Luke is really out of his element tend to be against pure Force users like Joruus C'baoth and Exar Kun's spirit. As long as Luke can make an actual fight of it, there's pretty much no combat scenario he can land in where he won't at least hold his own fairly well.