Not really. Picture if there was a glowing white-hot anvil. If you ReTe the anvil over Vibria's head and release the spell (and succeed on your Aiming roll), does the fact that she has Greater Immunity to Fire protect her from going ker-splat?
Ice is just ice. There's nothing TO ice besides the ice. So no, being immune to fire doesn't protect you from being crushed by a hot anvil, but if you're hurled into the sun, you don't burn up. You'll still suffocate because there's no atmosphere, but you won't burn.
Okay, that makes sense. So, the ice dagger does nothing, because it's just basically solid cold.
On the subject:
If you're immune to fire, does molten lava harm you?
If you're immune to poison, is that just ingested poison, or does it also include poisonous gases? Such as, say, smoke?
Should immunity to fire also include immunity to smoke inhalation? (Carbon monoxide kills you even if your lungs don't burn.)
If you're immune to arrows shouldn't the impact still hurt you even if the arrows don't pierce your skin? (like getting shot with a bulletproof vest... you can still break a rib)
It's often difficult for me to figure out where to draw the line between "this doesn't make sense" and "this isn't mythic enough."
I do agree. As a Major Virtue, a Greater Immunity to Cold & Frost that cold damage and discomfort is negated, and items made of frost (ice) cannot ham you. You can still drown in icy waters, but you cannot be suffocated by being buried in snow. An ice dagger will not harm you, but a super chilled frosted over steel dagger can still kill you.
I agree with Peregrine that an immunity to Cold should not protect you from an Ice dagger. It's not the cold that kills you, it's the piercing.
Likewise, humans may be "immune to kryptonite", but you can still smah their heads with a block of it.
However, a Greater Immunity to Cold and Ice will protect you, so it's mostly a question of write up 
If you're immune to fire, does molten lava harm you?
Hard one, since, by RAW, lava is mostly Terram and Aquam. Probably not, although I can see it going either way.
If you're immune to poison, is that just ingested poison, or does it also include poisonous gases? Such as, say, smoke?
I don't think so, for a lesser focus. Yes for a greater one. Not for smoke (this would be more like deprivation). Poisons should more be considered as "can you create it in a lab, store it and sell it"
Should immunity to fire also include immunity to smoke inhalation? (Carbon monoxide kills you even if your lungs don't burn.)
No, but I tend to think that this can often be ignored, save in the most realistic RPGs or in serious situations. I can very well see a character with both Immunity to Fire and a Death Prophecy (Fire Shall be your Undoing) dying that way. I tend to think that this could very well be a bad way for a GM to pester a player. How to put it? If fire is the primary danger, all players should suffer the effects of fire, period, and forget the smoke. If the smoke is the primary danger, then all players should suffer it.
If you're immune to arrows shouldn't the impact still hurt you even if the arrows don't pierce your skin?
No.
Do we get to use both adventure xp and advancement from other sources (e.g. study totals or exposure from lab work), or is it only one source, period?
only one source per season
I always just make sure to do lab work in a season where I've gotten adventure exp.
You can still do lab work this season. You just cannot gain exposure xp if you are already taking adventure xp this season.
Also, if somehow you manage multiple adventures in a season, you can take xp from all of them. All combined adventures constitute one single source.
In the library, I allow you to study more than one book in a season. You can, say, split your study time between two diffenent books, dividing your study total for each in half (round down). This still counts as one source (the library)
How does this work for companions? Specifically, can they 'assign' one of their mandatory exposure seasons to an adventure season, take the adventure xp instead, and still get two 'free' seasons that year?
Mission Improbable is on the planner for being in Spring, which is when Vibria was going to try to scribe some books from Sigmundo's incredibly drool-worthy collection. So, she would only have gotten 2xp Exposure to Scribe, anyway. I was just checking more for future reference than anything else.
Not exactly sure what you mean. Please tell me what you specifically want to do.
In Spring, you take the adventure xp. Everyone got 10xp, magi and companions and grogs alike. It was a long drawn out story compressed into a timeframe of only about a week of game time.
Whatever you do with the rest of the season is whatever you do. Whatever that may be. You can't get any extra xp from it though.
For seasons thereafter, I think companion characters should plot out activities like magi do, you do this in summer and that in fall. Whatever it may be.
For grogs, just give them the abstract 5xp per season and don't worry about details unless you want to (keep in mind that carefully manicured grogs are eligible to one day advance to being companions according to the Grogs book).
For familiars, they only get xp for doing things with you. They get the adventure xp, exposure xp from the lab, and can be taught by you. But they cannot study something else while you study, and they cannot study the same book as you (maybe if you had two copies of the same book, but let us not open that can of worms).
As a companion, Bernhard has two free seasons per year to study or practice or whatever. The other two seasons he works for the covenant or for Vulcanus and gains exposure xp. In 1230, he gains adventure xp in Spring for Mission Improbable. I want to designate that season as one in which he's working for the covenant, therefore using two of the remaining seasons as his free seasons. I think that's allowed, based on the rest of your response.
I swapped around two of Vulc's seasons (making Vibria's chainmail an Item of Quality, and enchanting it) so that the season that I used Bernhard's assistance in the lab was the same season he got Adventure xp. I figure that's not too metagamey since Vulc knew Bernhard was going out in the spring and would have planned for when using his assistance least impacts Bernhard's own stuff.
As a companion, all your time is free. You simply have an arrangement to work two seasons a year. And going on adventure on behalf of the covenant counts as work IMO. So you are good.
That works too
The effect must be identical in every way, including all scores and parameters (uses per day, Penetration, etcetera), and the utilization of a Form & Effect bonus for Shape & Material
It can only be used for your Talisman, and is useless for anyone else. Though it may seem useless, if your talisman is ever destroyed, the text makes it easier to recreate.
No. It can only be used by the original magus, and only for his own talisman.
Same shape & material for form and effect is essential. However, if your Lab Total did not bother with any form & effect bonus, the shape & material becomes irrelevant and you can put that effect into just about anything.
That means my item creation lab texts are useless to other magi, because Vulc always uses runes 
No point in scribing them, since I can recreate my own enchantments from untranslated texts.