The original sentence is fine. The pronoun 'they' is the subject of the second part of the sentence, and it refers to the tickets. The word 'cost' is a verb here: 'They cost even less than the airport tax.' Your suggested sentence "Nowadays, some airline tickets are so cheap that the cost even less than the airport tax." is incorrect grammatically because the second half has no verb. '... the cost ( ) even less than the airport tax' makes no sense. If you are using 'cost' as a noun, you need to have a verb (is) following it. The form 'their's cost' is simply wrong. This form doesn't exist. When you see correct English in books you should accept it and learn from it - not try to change it.

"nowadays, some airline tickets are so cheap, that they cost even less than the airport tax. "nowadays, some airline tickets are so cheap that their cost is even less than the airport tax. I think it's the same :)

"The" is the definite article. You can ONLY use it before a noun. "They" (它们) is a pronoun. It takes the place of a noun. In this case, "they" takes the place of "airline tickets".