To answer your first question – it depends. Sometime matches are true matches and whoever is providing the matching funds will only give the amount to be matched. So in your scenario, the organization would only get $800,000 in matching funds. This is how employer matching gift programs run, and there are other third parties that provided this type of match.
In other situations, the match is more of a challenge grant where the person or organization providing the match will still give the full amount (or may have already made the gift). What often happens in these situations is that your specific gift may direct the matching funds to a specific program at the organization. For example, I could provide (theoretically) a challenge of up to $100,000 to a college that will match donations, and if one donor gifts $10,000 to the English department and one donor gives $10,000 to the Math department, each department would get an additional $10,000 from the challenge fund. If no one else gave, the remaining $80,000 would go to wherever I specified.
For your questions about cash versus stock, I’ve never seen a match that wouldn’t match both.