Learn Dutch: Eindelijk komt de bus. - Finally the bus is coming.

Because the sentence starts with the adverb Eindelijk (Finally). In Dutch, when a regular main clause starts with anything other than the subject, the verb and subject swap places to keep the verb in the second position. This rule is called inversion.

The English translation says 'is coming', but komt is just 'comes'. Why not use a continuous form?

Dutch doesn't use continuous forms nearly as much as English does. For short actions or movement like a bus arriving, the simple present tense (komt) is the most natural way to say 'is coming'.

Since the bus is late, how would this sentence change if we used the time particle pas instead?

If you said De bus komt pas, it means 'The bus is only arriving now.' While pas emphasizes the delay and the fact that it didn't happen sooner, eindelijk focuses on the emotional relief that the wait is over ('At last!').

Is eindelijk the word I should always use for 'finally'?

Eindelijk is specifically for expressing 'at last' after a long wait. If you are listing points and want to say 'finally' to mean 'lastly' or 'in conclusion', you would use ten slotte instead.