First, graphite is a conductor. It is used in the umbrella because it is a strong material that resists bending. Second, a lightning strike makes everything a conductor, including people and trees.
In order:
- No. The primary target of lightning is the tallest object in the vicinity. Trees are very commonly struck, despite being non-conductive.
- There is possibly a slightly greater chance of being hit with an umbrella versus not having one due to the height difference. The greater danger is the umbrella slowing down you running for shelter.
- If you are holding it, yes, no matter what the umbrella is made of. Millions of volts make everything a conductor.
To stay safe in a thunderstorm consult here: http://www.lightningsafety.noaa.gov/tips.shtmlhttps://www.weather.gov/safety/lightning