I think that these are both possible, though the second one requires a little more work.
#1 - There should be a URL created for online giving when you set up your Weblink Online Giving. You can use this URL behind the scenes for the button to direct people to F1's online giving.
#2 - While there may be a better way to handle this, the best way I can think of off the top of my head is to set up a sub-fund for each person. That way the giver can give towards the "Mission Trip" Fund and choose the appropriate person from the Sub-Fund. If that doesn't help, I'm going to have to defer to someone with more knowledge.
In Him,
-Peter Schott