what it says on the tin. Israel never withdraws unilaterally and hands Hizballah the win.
Ehud Barak promised to withdraw for Lebanon.
Was his electoral mandate in 1999 primarily about withdrawing from Lebanon or something else, perhaps domestic?
Did he have an electoral mandate to try out a two-state solution? Yes or no.
Did he have an electoral mandate to try out a two-state solution? Yes or no.
Even if he did have a mandate to try it out it was obviously revocable after the rejection of his proposals, the Sharon visit to Temple Mount/Al-Aqsa and the 2nd Intifadeh.
Omar04 beautifully addresses the matter of Lebanon, but to answer the second part of your question - he definitely had an electoral mandate towards peace with the Palestinians, including possibly some type of two-state solution. A large part of why Meretz was going to be in coalition with him (before he flipped on them for Shas) was because of his stated commitment to peace negotiations. Indeed, he met with Arafat in 2000 at Camp David as part of the ongoing post-Oslo peace process. Camp David was successful enough to spawn a sequel despite the Intifada (the 2001 Taba Summit), which obviously had no follow-up after Sharon won in 2001
After the Second Intifada, though, the Israeli public was not in a mood for peace. They saw themselves as being taken advantage of, with Israel being willing to discuss peace even as the Palestinians started a "completely unjustified" violent uprising. And this is shown in the results of the 2001 election, where the somewhat dovish Barak lost to the very hawk-y Netanyahu.
What's interesting is that now that I'm thinking about the two events as being chronologically similar, the Israeli public might well have been much less enthusiastic about leaving Lebanon after an Intifada.
Did the unilateral non-negotiated withdrawal of Israel from Lebanon encourage Palestinians to overestimate what they could get from an uprising?
That's a very good question. It seems like a good theory, but I don't know. Yasser Arafat wasn't a stupid man and he would have realized after the first couple years that it wasn't working, and yet he kept the Intifada going until his death.
The Intifada was more or less out of the PA hands. Arafat did his damnest to put it down, Israeli propaganda (swallowed wholesale by the Americans and Euros) nothwithstanding.