I don't think the phrase "Hakuna Matata" is neither an "English" nor a "Latin" expression. Then, where did it come from?

What does “lay low” mean?

To lay low means to keep a low profile (stay quiet) or to not draw attention to oneself. A similar expression is "stay lowkey." Ex: I got into a fight with my parents, so I'm going to stay lowkey for a while. Ex: The police put out an arrest warrant for the criminal, so he decided to lay low for the time being.

What does "this thing" in the lyrics refer to?

"This thing" can refer to a habit one has, usually shown in past experiences. So here, she's saying that she has this constant habit, proven by past experience and situations, where she gets older but not wiser! Ex: I have this thing where I bite my nails when I'm nervous. Ex: She has this thing where she doesn't respond to anyone when she's overwhelmed.

Why "say" is present form in here?

"Say" is in the present tense here because the entire sentence is in the present progressive tense with the verb "twist" + ing. Since he is talking about something which has happened and continues to happen, he can use the present tense form "say", even though he has already "said" something.

Well I guess "[noun]+free" is used quite often to refer something doesn't contain any [noun], right? Could you give me some other examples?

Yes, that's right! This is mostly used with foods. Some other common ones are "colorant-free," "sugar-free," "dairy-free," "wheat-free," and "egg-free." Ex: This custard is egg-free since you have an egg allergy. Ex: I'm dairy intolerant. Is this dairy-free?

Who does "as" refer to?

"As" in this case is a continuation of the previous sentence and is not beginning a new sentence, there is just a pause in between. So the full sentence here is "they are best known for their traditional role as fire dogs." "As" here is referring to the Dalmations being fire dogs.