Can I use cookies to access a membership area

Greetings -
I have two questions related to visitors' access to protected content on my web site.

1. Can I have the box ticked by default so members remain logged in between sessions?

2. Is it possible to have the Memership2 plugin admit someone to a given level on my site if they have a cookie set in their browser from another website? More specifically, I'm trying to set up my site to work with another site, as follows:
- strangers could visit my site and read some content, but they'd have to register to read one level deeper into my site.
- once they have access to that first level, they could then register at a second, sister site. Those who had registered at both sites would have access to levels one and two on my site. Access to level two would only be possible if they were logged into the second site, and had a cookie on their computer as evidence they'd logged in.
- finally, if they pay for premium membership on the second site, they would have access to a third level on my site - meaning they could access any content on the site. That access would be made possible if they had logged into the second site as a premium member, and had a 'premium' cookie set on their computer.

Thanks for any enlightenment -

Dennis