I can't answer all of these, because I dont really know for some of them but for the rest
1. There's a couple reasons, for one, the sword is a Lost Thing, for another, the sword still had the Grimbeard's will/map to the Jewel inside.
Leaving it with Hiccup is a risk, better to get it out of the way, just in case.
2. It could've just been an additional sign of sorts, but also might've been sorta made up? It's kind of unclear.
Excellinor was the one that suggested the swordfighting competition be the decider, probably because a Swordfighting Competition is more likely to get the Vikings to agree to the whole King Thing, and because Alvin had been practicing with Flashburn, so it might not strictly have been what Grimbeard meant in the first place.
It could've also just been a 2 part process, like, The Winner will be King, so you'll accept this and then go to the Official Ceremony and back them up.
Also like, Hiccup did win, even though he got disqualified, so in the whole Prophecy/Future Telling thing, Grimbeard might've put it it like "Hey, this guy will be my heir... you wont pay attention to that cause you're idiots so here's the other things that'll happen to prove this" lmao
3. Yes, he likely guessed it was vorpent venom, and he could have fought her, but I highly doubt Fishlegs and Camicazi would let him fight her on his own, and he probably didn't want to risk it not being Vorpent Venom and potentially getting them all killed.
He also had no way to know for certain that Stoick was actually in that cabinet, so if Hiccup does fight her and he's not in there, they're back to square one.
He needs the crown anyways, it's best just to go with it for now instead of risking it.
4. I dont know, I've never quite understood the Liars Dice thing lmao, someone explained it to me once but I don't remember.
As for the rules, Cressida didn't make this game up, it's an actual game thats been around for ages
5. The timeline you have listed is incorrect, some of the books have errors in the listed dates, so every so often people get really confused by it.
It's not 6 months between books 9 and 10, it's about a year
and its not 2 months between books 10 - 11, its about 1 month.
I say about because how often do we say 'a year' or 'a month' and mean exactly 1 whole year or month
Doomsday takes place Midwinter, which lines up with those dates.
As for why the assumption that Doomsday is the 31st, i think its because the full name it The Doomsday of Yule, and Yule...
...is right around that time
Technically speaking, there's also this
Which that second date would actually fit the given timeline better (about 1 year from the last new year in book 10, then about a month later in book 11) but I think End of December might've also been picked because of the 12 Days of Christmas.
6. Winning the competetion wasnt strictly necessary to be King, but like i said for #2, i think it was sorta both a 2 step process and about the Vikings being willing to accept the person as King
The Druid Guardian doesn't need to say anything about the Swordfighting Competetion because its not The Decider™️, it's not what his job is, it's not the most important thing.
But he does mention Being the Chosen Representitive of the Archipelago as a thing, so again even though everyone agreeing isn't necessary, I think it's all kind of in that same vein, ya know?
Anyways, yeah 👍hope that helps somewhat lol
I got the sleepies today and I'm having trouble remembering things right off, so if anyone sees something I got wrong, please feel free to correct it