Making intention of sending someone for umrah if medical reports are clear

Q: Few months back my father's prostate report came abnormal. I made niyat to Allah Ta'ala that I will send someone for umrah and will stop watching TV and movies if his next report comes normal. Then I thought that sending someone for umrah is a good act and stop watching Tv and movies is a bad act, so I made niyat of it irrespective of the result of his next report.

My question is, is my oath a formal oath? Because I made niyat like this "I am making this niyat that I will send someone for umrah and stop watching Tv and movies from now.

According to what I have learnt, an oath or niyat should be taken with the name of Allah like "in the name of Allah" or "wallahi" or similar to this? I am 99 percent sure that while making the niyat I have not taken the name of Allah.

Note: Alhamdulillah with the blessing of Allah I sent one person for umrah but I am unable to avoid TV and movies, i just want to make some changes in it.

Are hair roots impure?

Q: Is the root of the hair najis? And if so, what does this really mean? Since the hair that the body has naturally shed has the root intact (as opposed to cut hair), do I then assume that all hair that I see on the floor or in the bathroom has a part of it that is najis?

As a consequence, if I am eating soup or drinking water, and some hair falls in, does that make the entire liquid najis? If some fallen hair on the floor sticks to my wet foot, do I have to wash my foot? How exactly is the najasat of the hair root transferred?