A lány vajat is tesz a szendvicsbe, miután hazaér.

Because -t is the most common accusative (direct object) marker in Hungarian.

• vaj = butter (dictionary form)
• vajat = butter (as the thing being put/added)

So in vajat tesz, the butter is the direct object of tesz.

is means also / too / as well. It usually goes right after the word (or phrase) it focuses on.

• vajat is tesz = she adds butter too (in addition to something else)
  If you moved it, the emphasis would change. For example:
• A lány is tesz vajat... = the girl also adds butter (someone else does too)

Hungarian marks location/direction with case endings. -be means into (movement into something).

• szendvics = sandwich
• szendvicsbe = into the sandwich (i.e., into the sandwich as a whole / inside it, between the slices)

If you mean “onto/over the surface,” Hungarian often uses different endings (depending on the exact meaning), but sandwiches are commonly treated as something you put fillings into.

Because the sentence treats butter as an indefinite/non-specific amount: “(some) butter.” Hungarian often omits an article in this kind of usage.
If you wanted to emphasize a specific butter, you could say:

• A lány a vajat is a szendvicsbe teszi... = she puts the butter into the sandwich too
  That version also tends to trigger different verb conjugation (see next question).

Hungarian verbs have an indefinite vs definite conjugation.

• tesz = 3rd person singular, indefinite conjugation
• teszi = 3rd person singular, definite conjugation

You typically use the definite form when the direct object is definite (e.g., has a/az, a possessive, a pronoun, etc.). Here the object is vajat (no definite marker), so tesz is natural.

Yes—Hungarian word order is flexible, but it changes focus/emphasis. The neutral version here is:

• A lány vajat is tesz a szendvicsbe, miután hazaér.

For example:

• Miután hazaér, a lány vajat is tesz a szendvicsbe. = same meaning, but the time clause is foregrounded first.
• A szendvicsbe tesz vajat is... = focuses more on the destination (into the sandwich) as the important part.

Because miután hazaér is a subordinate clause (“after she gets home”), and Hungarian typically separates such clauses with a comma, similar to English:

• “..., after she gets home.”

miután is a conjunction meaning after + a full clause. It introduces an action that happens later than another action.

• miután hazaér = after she gets home

után is more like a postposition meaning after + a noun/phrase:

• a vacsora után = after dinner
  You generally don’t use után directly with a finite verb the way miután works.

hazaér is a common Hungarian verb meaning arrive home. It’s built from:

• haza- = home (as a directional verbal prefix: “to home”)
• ér = arrive/reach

Hungarian often forms verbs with such prefixes, and they’re usually written together (though in some constructions the prefix can separate).