Assuming no one contested registration of the trademark within 5 years, is the trademark valid?

Assuming no one contested registration of the trademark within 5 years, is the trademark valid?

section-1 500 words.

Question I – The year is 2012 and the light bulb has not yet been invented.  Thomas Edison and Nikola Tesla are in a bitter feud to see who can bring electricity to the masses first.  In this alternative history, it is indisputable that Tesla invented the lightbulb first.  However, Edison filed his lightbulb patent prior to Tesla.  According to the law in 2012, who holds the patent to the lightbulb?

Question II – In May of 2013, Jesse plans to open a law office named “Peace Law” and uses the peace symbol as part of the logo for the business.  That same month Jesse registers the trademark with the Patent and Trademark Office.  Jesse opens the law office for practice in July 2015.  Assuming no one contested registration of the trademark within 5 years, is the trademark valid?