Correct. They are paid by U.S. importers, who either eat the costs or pass them on to consumers - or both.
Well a trade war is coming, most likely. And I say that because in the case of Canada and Mexico, we have now twice violated two free trade agreements (under Trump's direction). Sorry, but you can't break a decades-old free-trade pact and not expect our trading partners to accept it. They will use their own leverage to retaliate. Justin Trudeau has already conceded that Canada is in for some economic pain but they are not rolling over -- and his political rivals, the conservatives, have already supported his position. There's political solidarity in Canada against the United States because this was completely unprovoked. Canada has accepted that it is, to a degree, fucked. Now it wants to fuck us back, and they seem ready to do so.
You're talking about two different things. Tariffs can't do anything about the national debt -- that's fiscal policy, not trade policy. Tariffs are about the balance of trade between two countries. Here's the really ironic thing about tariffs: if tariffs in the United States are more effective than we bargained for, and if they really hurt Canada's and Mexico's economies, we might end up with even a worse trade imbalance. Thus, there is a scenario in which tariffs actually backfire. That's because if our tariffs hurt Canada and Mexico's economies, we make their capital inflows (investment) less attractive and our foreign investment more attractive, which strengthens the US dollar and weakens their currencies.