In get started and get lost, why is get followed by the past participle? I don't understand it. Is the past participle an adjective?
In these expressions, "started" and "lost" are not verbs. Get is the verb, (transitive) and it is a command, with the second person understood as the message receiver. But what about the fake verbs? They look like adjectives, as you guessed, but I think they are nouns. Each is a condition, and not a modifier of a noun.
JamesK said earlier these are "fixed idioms" which means they are customary expressions that exist as an ensemble. Such expressions often don't have a literal translation, you must learn them as if they are equivalent to a word. I know some French and Spanish; these languages have plenty of fixed idioms that are untranslatable from their actual wording.
This is sometimes called the "get" passive form.
The normal passive is be + p.p. "The apple was eaten". But in casual use, and particularly when speaking of events that are bad we might use "The apple got eaten."
Some particular expressions use the "get" passive: you have mentioned two of them "get started" and "get lost". You can treat these as fixed idioms.