Learn Hardcore Romanian: Totuși, biletele ieftine apar uneori înainte de vacanță. - However, cheap tickets sometimes appear before the vacation.

Totuși is an adverb/conjunction meaning however or nevertheless. It introduces a contrast with what was said before. When placed at the beginning of a sentence, it’s set off by a comma to signal a brief pause:
Totuși, …

Can I use dar instead of totuși in this sentence?

dar is the conjunction but, used to join two clauses (e.g. „Am vrut să merg, dar nu am avut timp.”). Both dar and totuși express contrast, but totuși can comfortably start a sentence in both spoken and written Romanian, while dar at the very beginning is more colloquial.

Why is ieftine placed after biletele? In English adjectives come before nouns.

Romanian adjectives normally follow the noun they modify. So biletele ieftine is literally tickets cheap. You can front an adjective for emphasis or a poetic effect, but the default order is noun + adjective.

Why does biletele have the suffix -le, and how would I say “cheap tickets” in an indefinite sense?

The suffix -le marks the plural definite article: biletele = the tickets. To say “cheap tickets” indefinitely, add niște (some):
niște bilete ieftine

What does apar mean, and how is a apărea conjugated in the present tense?

apar is the 3rd person plural present of a apărea (to appear/to show up), so it means they appear. Present conjugation:
eu apar
tu apari
el/ea apare
noi apărem
voi apăreți
ei/ele apar

What does uneori mean, and can its position in the sentence change?

uneori means sometimes. Romanian word order is flexible for adverbs, so you can place uneori before the verb, after the verb, or at the start of the sentence:
– Totuși, biletele ieftine apar uneori înainte de vacanță.
– Totuși, uneori biletele ieftine apar înainte de vacanță.

Why do we say înainte de vacanță? What’s the function of de here?

înainte de is a compound preposition meaning before. When you specify what comes before (vacation), you must use de plus the noun. Dropping de (înainte vacanță) would be ungrammatical.

Why isn’t vacanță inflected for case after de, and why is there no article?

Romanian nouns after prepositions remain in the nominative/accusative form and don’t change their endings. Definiteness is marked by the article: here vacanță is indefinite, so no suffix. If you wanted it definite you’d say înainte de vacanța (though in practice you’d often specify further: înainte de vacanța de vară).