I can't remember why this is byddwch and not fyddwch. Is it because of the question word at the start of the sentence?
It's a good question. You know a declarative statement has no mutation and a question has a soft mutation for starters.
When it comes to question words, a good guide to spoken Welsh is that the pronouns, pwy (who), beth (what) and faint (how much/many) are followed by a soft mutation.
The others, the adverbs, have no mutation: ble (where), pryd (when), pam (why), sut (how), faint o'r gloch (what time), anything that starts with a preposition etc.
This is a guideline. You'll find in more formal Welsh that there will be fewer mutations after the pronouns, and in informal Welsh there may be mutations after all question words. There are actually quite a few irregularities in this Duolingo course. So stick to the above and you should be fine.
Byddwch is the base word and can be used even in a statement "Byddwch chi'n mynd i'r siop." (You will go to the shop. It can sometimes be "fyddwch" since traditional it would be preceded by a "Mi" i.e "Mi fyddwch chi'n mynd i'r siop". I believe that it will always be "Byddwch" after "sut" since it would never had had a "Mi" before it.
It would if it was a question that once had an "a" interrogative particle before it, but since the question starts with "Sut" I don't believe it would have. This is just what I feel is correct so you might want to check with one of the contributors.
Some questions words cause a mutation, some don't.
With simple questions, Pwy?, Pa?, Faint?, Beth? and Sut? (what sort of) do. Sut? (how), Pryd?, Ble? and Pam? don't. (But in the wild, some people do use a mutation after Pryd? and Ble?.)
Sut gwnaethoch chi gyrraedd Llundain? (How do you get to London?)
Sut ddyn ydy Siôn? (What sort of man is Siôn?)
Pryd bydd y cyfarfod, ble bydd e, beth fydd ar yr agenda? (When will the meeting be, where will it be, what will be on the agenda?)