Of course they do. Libertarians in general believe that private charity should replace the government safety nets. Private charity is the first response most libertarians will give to the question of what do we do with the indigent.
You're confusing libertarians with Ebenezer Scrooge.
The ones I have come across have generally stated that they will engage in private charity if the government stops taxing them. It holds all the value of an election promise. But if you're not on the poverty line and you're not already giving to charity, you won't magically become charitable with the application of more money. It's just not how people work.
And certainly, if you're not giving to charity now out of some sense of 'cave deservedness' like clarky07 above, then nothing will make you give continuously to people in lifelong need of financial support.
Yes, they do. For libertarians the big sticking point isn't this or that thing should be done. It's whether or not people have the social obligation to help the poor or the legal obligation.
"Those who think government should do more to redistribute income are less likely to give to charitable causes, and those who believe the government has less of a role to play in income redistribution tend to give more"
...they do?