Absolutely not, and I have no idea why people think it does. It's like there's lithium in the name, so people instantly assume that it makes up a substantial portion of the cost. It does not.
Lithium is cheap, currently lithium hydroxide is at ~30$/kg. Of the total cost of a typical LFP battery, the lithium is ~2%. The rest of the materials round up to ~10%, the bulk of the cost is capital and labor.
I don't think so, but I don't have specifics. There's some sources that a typical Tesla battery pack might have about 10-12 or so kg of lithium. LFP is a different chemistry, so it might have a different proportion of lithium but it's probably not hugely different. Lithium isn't particularly expensive (though it might become expensive in the future if demand goes up and production does not).