With multiple monitors, can I pick which one an application will load on

in Spaces Preferences you can set the browser application to appear on all spaces. that's how i have my Terminal set up, among a few others. unless you move more apps onto your secondary monitor, the browser window should constanly stay visible even when switching around Spaces.

This is an excerpt of Appple's Application Kit Release Note for Lion regarding fills screen apps & multiple desktops.

Multiple Monitors and Full Screen (New since early 2011 seed)

We have made some simplifying assumptions for Lion Full Screen behavior on multiple monitors. Multiple monitors are treated as a single unit by Spaces, and therefore are also treated as a single unit in Full Screen. This means that all monitors will be dedicated to windows belonging to the full screen application, and there can be only one primary full screen window visible at a time. A secondary monitor is useful for inspector windows.

Secondly, because the menu bar is located on the main monitor, the primary full screen window will be located on the main monitor as well. This allows the menu bar, floating toolbar, and full screen window to maintain their interrelationship on the same monitor.

full document can be found here: http://developer.apple.com/library/mac/#releasenotes/Cocoa/AppKit.html

According to this document it isn't possible.