Learn Hardcore Japanese: asita syatyou no mukai no seki ni suwaru yotei desu. - Tomorrow I plan to sit in the seat opposite the president.

社長の向かい means “the spot opposite the president” (first の connects 社長 and 向かい).
Adding の席 then gives “the seat at that spot” (second の connects 社長の向かい and 席).

What does 向かい mean here, and how do I use it?

向かい is a noun meaning “opposite side” or “across from.” To say “across from X,” you use Xの向かい. To modify another noun (like 席), it becomes Xの向かいのY = “the Y across from X.”

Why is there no particle after 明日 at the beginning? Shouldn’t it be 明日に or 明日は?

Time words at the start of a sentence often act adverbially and can drop particles. Simply 明日 means “tomorrow.” If you say 明日は, you explicitly mark “as for tomorrow…,” but it’s optional here.

What role does に play in 席に座る?

The particle に marks a location where an action occurs. 席に座る literally means “to sit at/on a seat.” Without に, the location relationship is unclear.

Why is 座る in its dictionary form before 予定です, instead of polite 座ります?

The construction [dictionary-form verb] + 予定です is a fixed pattern meaning “plan to [verb].” You always use the verb’s plain (dictionary) form, then add 予定です to keep the sentence polite.

What’s the difference between 予定です and つもりです when talking about plans?

予定です implies a scheduled or arranged plan (often set by someone or in a timetable).
つもりです expresses personal intention or will (“I intend to…”).
Both can translate as “I plan to…,” but 予定 feels more like a confirmed arrangement.

Why is the subject (like “I”) omitted in this sentence?

Japanese often drops subjects when context makes them clear. Here, it’s understood you’re talking about yourself, so 私は is unnecessary. You could add it for emphasis:
私は明日社長の向かいの席に座る予定です。

How would I make this sentence more casual?

In casual speech you could say:

明日社長の向かいの席に座る予定だ。
Or using つもり: 明日社長の向かいの席に座るつもりだ。