You sayin' Pepsiman ain't patriotic AS FUCK in Japan?

Is it Claws? Or Clause?

Judging by the kana, it looks like Claws, though it could be Clause (as in, scientific clause/law). But multiple people on Reddit say it's Cross.

Should we wait till the Rider's name is said in-show or is at least readable?

TV Asahi website has the official name as Kamen Rider Cross

Yes. It doesn't even match the ku-ro-zu kana.
Japan weird names consistency again.

Hmm. That's weird. Like, really weird.

Is this gonna be Pallad/Parad/Parado all over again?

No.
Parad case is more like the BloodStark/BloodStalk one.
Kurozu katakana into Cross is even weirder.

That's really weird, then.

Do we go by katakana, by the show, or by the eventual Ganbarizing card?

Anyone think this KR Grease will be from Hokuto? I so wish he would be a genius like Sento too.

Kamen Rider Cross/Claws/Clause/Close/Clothes/Cloud/Christopher Robin, I'm not sure what this guy's name is anymore :lol

Anyone know where I can find some translations, my Japanese is a little rusty/non existant.

Here's something.

Though as has been pointed since, the name of the Driver/Jellies is probably Sclash rather than Scrash, which I used. Also I think the Twin Breaker was Sougeki Souten rather than Souka Souten. And the sound for two bottles/jellies is "Twin Break," not "Double Break" like I put there.

edit: actually turns out I screwed up the Sclash sounds too. Should be "Tsubureru! Nagareru! Afurederu!" and "Tsuburenai!" for regular bottles. The first part of the Rogue jingle is "Wareru!" So that was my bad.