For example: After seeing Banefire and choosing to cast it, would the new top of my library be available for viewing before I choose an X value? If not, when is the soonest I can check the next card?
You cannot look at the top of your library while casting a spell like Banefire from the top of your library. You can look at the top of your library as soon as you have finished casting the spell. This is specified in rule 401.5:
Some effects tell a player to play with the top card of their library revealed, or say that a player may look at the top card of their library. If the top card of the player’s library changes while a spell is being cast, the new top card won’t be revealed and can’t be looked at until the spell becomes cast (see rule 601.2i). The same is true with relation to an ability being activated.
Since the first step of casting a spell, listed as rule 601.2a, is
To propose the casting of a spell, a player first moves that card (or that copy of a card) from where it is to the stack. [...]
Casting a spell from the top of your library counts as the top card of the library changing while the spell is being cast.
The final step of casting a spell, rule 601.2i in the same section, says in part
Once the steps described in 601.2a-h are completed, effects that modify the characteristics of the spell as it's cast are applied, then the spell becomes cast.
Once you complete the entire process of casting a spell, then the spell becomes cast and you can look at the card on top of your library.
You have to wait until you finish casting Banefire (which includes picking an X value and paying its cost) before you can look at the (new) top card of your library.
If the top card of your library changes while you’re casting a spell or activating an ability, you can’t look at the new top card until you finish casting that spell or activating that ability. This means that if you cast the top card of your library, you can’t look at the next one until you’re done paying for that spell.