Old Saybrook

Amtrak regional trains often stop at Newark airport on their way to Washington DC and points south. Or one can change at NY Penn Station to New Jersey Transit. Either way works.

Yes go to the amtrak app. More times available from New Haven though. Usually more convenient. Not every train stops in OS.

at Saybrook,does the $10/car cover days of parking?

Yes, inside the station

There is no taxi stand, so you would need to call for a pickup.

Amtrak will take you from Saybrook to Bridgeport but you have to book 14 days in advance for prices to be reasonable!

There are no cars available there

Yes it is marked, you will see signs on the light posts and different colored spots. I do fine however that they are always full, I wish Amtrak could better control it as I am sure not all of them are Amtrak riders. Do note the spots right next to the station are daily pair spots ($10 a day). To the west of the station there is new CT DOT parking that is usually open and without a cost.

Yes

To renovate it.

Not that I've ever heard of, although they are rehabilating the freight line that goes North from NL, and that could be used in the future.

You should be able to take the Amtrak to Boston most days

The cheapest method is to take Metro-North from Grand Central to New Haven and change to Shoreline East. Total off-peak return adds up to around $50 per person.