Make sure you have "Allow Exposé, Dashboard and others to use the screen" in keynote preferences under Slideshow.
Then, before you slide over to Safari, press the "F" key to pause the slideshow. This prevents the slideshow from disappearing.
You can then slide over, or use Mission Control, etc without affecting the slideshow.
There is a bug here, to go back to the slideshow you have to slide back to it or click on the desktop in Mission Control; you can't click the Dock icon or Cmd-Tab back to it.
(Clicking the dock icon / cmd-tabbing will just hide the dock and menubar but not take you back to the presentation)
Also: When you get back to the slideshow, the first key press or mouse click unpauses the presentation, so you will have to click / press again to advance to the next slide.
Side note: instead of having keynote and Safari on separate desktops, you can have them both as fullscreen apps. This hides the Dock and Menubar, so when you show the web page, there is more room for it.
Staying with the keynote format, no. I you export to a QuickTime movie, you will have the best chance to preserve the slide deck as it was on the Mac when saving it alone doesn't work.
You can only work around this by getting the mac pixels on the iPad since this appears to be an inherent limitation of the iOS app as written today.
Have you tried using Air Display to add the iPad as a second monitor and tell Keynote to present on the iPad? The company does awesome work and like any screen sharing over a network, there is latency and/or slight rendering artifacts, but it's better than other apps in that space as any I've seen.
Best Answer
OS X has a command-line tool called screencapture that makes a screenshot. Try this:
screencapture -T 5 -t tiff ~/Desktop/screenshot.png(the "-T 5" option specifies a 5-second delay; the "-t tiff" option specifies saving the data in the uncompressed tiff image format)With a full-screen screenshot, you can use the crop tool in Preview.app to save the region of the screen that you want.