In the US environmental regulations, the cost of producing power, labor costs, would all drive up the price of the end product in a way that makes it totally noncompetitive. That's also why the US and some other countries are investing in other ways to find lithium (among other things) on seabeds, where it's hoped that extraction would be less expensive. Of course the threat to the seabed environment is a concern, which in turn might drive up prices by imposing regulation, etc etc etc.
In an export model, yes. However, given their negative externalities (including geo-political factors), importing countries may place tariffs on Chinese lithium in order to make use of other sources.
If the total embodied value of lithium in any particular product is small compared to the overall value of the product, the tariff might not represent a significant drag on the indigenous industry.
To be honest, the energy problem is more or less a solved problem with the current technologies we have. We just need to accelerate our pace of adoption to hard-reverse on fossil fuels (except Germany). We already have large reserves of Uranium, of which only a small amount is needed to fuel a power plant. We already have lithium battery tech to store the power. We already have solar panels being mass produced and adopted to fill in the gaps. All we need is connecting the dots and making sure these resources play well with each other in symbiosis.
I'm skeptical. China is already mass-producing batteries, securing as much lithium as possible. Additionally, US regulations will significantly increase costs for battery manufacturers.
So yea desert sand is essentially free, even if you pay for shipping.
It's also worth noting that Chinese prices are so low that certain tariffs can reach the stratosphere (eg: American 100% tariff on Chinese EVs), further making them unpopular with the commons.