Asalamu Alaikum Bro,
I hope you're well and in good state of Imaan, Inshallah.
Your question is a good indication of whats in your heart. Keep striving in the path of Allah. Nothing shall go in vain... Inshallah.
As others have said, it is recommended that you follow the sunnah of the prophet (sws) and try your best to be close to Allah. I have looked at the link you provide, it has mentioned some really good stuff...
Lets face it, you can't walk around in your everyday life with a thobe withouth making your life filled with obstacles... I'm somewhat familiar with your parents views on Islam... I'm not suggesting you don't wear it... but to find a balance.
So, what I suggest is that you buy a lose thobe that will fit over your clothes... that is what most of the people I know did... so, its easy to put it over your clothes when ur at the masjid and then take it off and go back to work or school or home...
As far as the cap is concerend... the prophet (sws) did both... he wore a hat and he didn't wear a hat... so either of them is ok.
Lastly, it is a sunnah, it is not obligatory upon you to complete it... but it is recommended because it brings you closer to Islam and softens your heart.
May Allah bless you and guide you through life. Ameen.
W'salaamz,
Hamid