Is boating permitted here (smaller kayak sized paddle operated)?

Yes, but you need a permit to do so. If caught without it you can be ticketed. I think its $30 for the year and that also gives you access to many other lakes in Harriman State Park

I love it nice place to be

Yes

I was told that you can bring in any boat as long as it is carried on top of your car. No boat trailers, no restriction on the of boat though except no powered board. (I specifically asked about very small sailing dinghy for older kids)

yed

Yes, but no paddle boarding. Not sure why.