1) In this instance, the word 'fast' is incorrectly used because it is an adjective, which can only describe the plane. What is needed instead is an adverb to modify the verb 'does', such as 'quickly' or 'soon'. The second problem with this sentence is 'after it starts to take off?' An airliner doesn't typically 'start to take off', it simply 'lifts-off', unless you wish to count also the time it spends accelerating down the runway. Typically, people mean to start counting time beginning with liff-off. This sentence might have been rewritten:
How soon does an airliner reach an altitude of 500m after it lifts-off?
2)The only problem I identify with this sentence is the misuse of the article 'the'. 'The' implies there is something special about an altitude of 30,000 feet. So in this case, I might rewrite the sentence on one of two ways:
What is the average time for an airplane to reach the cruising altitude of 30,000 feet?
since 30,000 feet is often the cruising altitude for planes. Or,
What is the average time for an airplane to reach an altitude of 30,000 feet?
in this case changing the definite article 'the' to the indefinite article 'an'.
The zero article typically is only used with proper nouns and prepositional phrases.
The English language has many subtleties that many native English-speakers often don't observe in its common usage, so despite these errors, most people will parse the meaning of either of these sentences without raising any objection. What this implies is that you too may relax from strict usage of the rules in common speech, but not so much in academic writing.