Is the nicotine that enters the body via medical patches for the purpose of quitting smoking prohibited? And is avoiding smokers, including one's brother, to prevent relapsing into smoking considered a severing of kinship ties?

You should have asked about the ruling on nicotine before using it, because it is forbidden (haram) due to the many intoxicants and harms it contains. Doctors have likened it to heroin and cocaine.

The Prophet, peace and blessings be upon him, said: "Indeed, Allah has made a cure for every disease, so seek treatment but do not seek treatment with what is unlawful (haram)," and "Indeed, Allah has not made the cure for my nation in what He has forbidden to them."

However, since you have used it, we hope that your repentance will expiate this sin.

As for maintaining kinship ties (silat al-rahim), it is obligatory, and severing them is forbidden (haram). Allah, the Exalted, says: "So would you perhaps, if you turned away, cause corruption on earth and sever your [ties of] kinship? Those are the ones whom Allah has cursed, so He made them deaf and blinded their vision." The Prophet, peace and blessings be upon him, said: "A severer of kinship ties will not enter Paradise."

Maintaining kinship ties is achieved through what is customarily considered a connection, such as contacting and giving gifts. You can choose a method of maintaining ties that does not conflict with your staying away from smokers, while striving to encourage your brother to quit smoking, by telling him that you cannot sit with him for fear of the harm of smoking, if that makes him consider quitting.