Où est-elle?-and the pronunciation of other liaisons you can find and the person who pronounced ""eSt-il"" can find the correct pronunciation here, one of the bests: http://www.languageguide.org/french/grammar/pronunciation/liaison.html-on this page there are only liaisons. This link is subordinated the link: http://www.languageguide.org/french/grammar/pronunciation/index.html
To be specific, in colloquial speech, one might say "Ton chapeau est doux?" (with the appropriate intonation,) but in standard French, there always has to be an inversion of the subject and the verb in questions. But, because things can't be simple, one can't say "Est ton chapeau doux?," so the stated subject stays in place and a placeholder subject pronoun is put in the typical inverted position, i.e. after the verb, which gives "Ton chapeau est-il doux?."
(Extra credit: Because of that, the [t] sound is always present in yes/no questions. That is to say, if you change "Ton fils mange du pain" into a question, it becomes "Ton fils mange-t-il du pain?", with an extra <t> to indicate the presence of that sound.)
ease your resentment over seemingly bizarre grammar with an outside link! voila: http://www.laits.utexas.edu/tex/gr/int3.html#pronoun (see second example)
What purpose does "il" serve in this sentence? I got it correct from just typing "Is your hat smooth" because I simply could not find any way to insert "it" or "he" into the sentence. so I just took a guess that payed off. I still just do not see what purpose it serves however. "Is your hat soft" why not "Ton chapeau est doux"? What would be wrong with that?
The "il" is used here so that we can ask a question using inversion.
You know that there are three ways to ask a question in French.
We can put "est-ce que" in front of a statement. "Est-ce que ton chapeau est doux"
We can use a rising intonation of the voice to turn a statement into a question. So as you suggest - "Ton chapeau est doux?" would also be correct.
Or we can use inversion. DL is showing use that in order to use inversion we have to use a pronoun - in this case "il".