Hello abengi, yes the paths are narrow and crowded in some places.
Are you far from parks? Usually the path/streets there are wider and
cars are off limits, most of the time. It looks like you just need some
more practice and you should be ok. I've been commuting in New York
City for 5 years now and I'm pretty comfortable on the streets and
paths. Contact me if you want a buddy to ride with, maybe give you
some pointers. I work nights though, so no 6:30am riding for me.
Afternoons during the week and weekends is fine. Riding thru Central
Park, Manhattan: