Uçaktan indikten sonra bagaj kontrolünden geçmek biraz zaman aldı.

Uçaktan = from the plane
indikten = root in- (from inmek, “to get off”) + verbal adjective -dik + ablative -ten
sonra = after
Together, Uçaktan indikten sonra literally means “after getting off the plane.” The pattern verb stem + ‑dik + ‑ten + sonra creates a time clause “after doing X.”

Yes. Uçaktan inince also means “when/once (we) get off the plane.”
• inince = inmek + -ince, a conjunction meaning “as soon as/when.”
• indikten sonra = “after (having) got off.”
Nuance: inince often implies immediacy (“as soon as we landed”), while indikten sonra is a more general sequence marker (“after we’d gotten off”).

We start with the compound noun bagaj kontrolü (“baggage control”). To add a case ending (-den) to a noun, Turkish first requires a 3rd person singular possessive suffix:
 kontrol + -ü → kontrolü
Then add the ablative -nden → kontrolü + -nden → kontrolünden
Result: bagaj kontrolünden = from/through the baggage control.

geçmek can be:
• Transitive – e.g. sınavı geçmek (“to pass an exam,” accusative)
• Intransitive – e.g. bir yerden geçmek (“to pass through a place,” ablative)
Here it’s intransitive (“to go through”), so we use the ablative kontrolünden, not an accusative.

biraz = a bit/some
zaman = time
almak (past tense aldı) = to take
Together, biraz zaman aldı literally means “it took a bit of time,” i.e. “it took some time.”

Yes:
• bir süre aldı = “it took a while,” slightly more formal/indefinite.
• vaktimizi aldı = “it took up our time,” emphasizing whose time was occupied.
By contrast, biraz zaman aldı is a neutral way to say “it just took a little time.”