Exactly! :)

"Kis" and "kicsi" are interchangeable when they stand directly before a noun, but not when they act as predicates -- only "kicsi" works in that role.

Alacsony means "not tall". While kicsi can be used for that too, its general meaning is "small (in all dimensions)".

Hanem is used to connect two statements that mutually exclude each other (and one of them has to be negated). Just like here -- a market can't be big and small at the same time.

Statements connected with de don't exclude each other by nature (and you can choose to negate one, both or none of the statements):

• A piac nem nagy, de olcsó. (The market is not big, but it's cheap.)
• A piac nagy, de olcsó. (The market is big, but it's cheap.)
• A piac nem nagy, de nem is olcsó. (The market is not big, but it isn't cheap either.)

If you want to express that the market is neither big, nor small, but somewhere in the middle, you use de.

• A piac nem nagy, de nem is kicsi. (The market is not big, nor small.)

Very good question!
Both can be used as an adjective in an adjective+noun kind of relationship. But "kis" is more common there:
a small building - egy kicsi ház / egy kis ház

Please note, "kicsi" may be over-represented in this course. At least for now, in the beta stage. In many of those sentences you may encounter, "kis" may be the more natural option.

"Kis" is used almost exclusively in compound words:
a small child - egy kisgyerek
a small dog - egy kiskutya

"Kicsi" is used exclusively in a noun+adjective type construction:
this house is small - ez a ház kicsi
you are small - te kicsi vagy
she is small - ő kicsi

So, "kis" cannot stand by itself, it always needs something that comes after it, something that it modifies.

Also, compare these two:
Kérek egy kicsi kenyeret - I want a small (loaf of) bread
Kérek egy kis kenyeret - I want some bread

Also, both can be made into an adverb, but differently:
"You arrived a little late" - "Egy kicsit későn érkeztél" or "Egy kissé későn érkeztél".