Can I share GAL's and calendars across two separate Exchange 2007 organizations that reside in separate forests? I can put a trust between the two forests.

GAL synchronization in pure Exchange 2007 environments wasn't supported until the release of Identity Lifecycle Manager 2007 Feature Pack 1. ILM 2007 FP1 provides a solution to synchronize the global address list (GAL) between two Active Directory forests in a Microsoft® Exchange Server 2007 Service Pack 1(SP1) infrastructure. The goal of a GAL synchronization solution is to synchronize users, groups, and contacts from one forest with contact objects to another forest. Microsoft publish document obout ILM: http://www.microsoft.com/download/en/details.aspx?displaylang=en&=tm&id=17887 Another options is set up resource forest. More you can find on http://www.msexchange.org/articles_tutorials/exchange-server-2007/planning-architecture/deploying-exchange-resource-forest-part1.html

Would the same thing work between and Exchange 2003 organization and an Exchange 2007 organization? I'm wondering if I even need to do an upgrade to get one organization at the same Exchange 2007 level.

Yes, you need to do an upgrade.