Well, yes, that's the point: it doesn't matter who pays up front, it affects the supply/demand equation, typically hurting everyone. Regardless of where I am in the chain, I'm not going to run at a loss and hope to make it up in volume.
It doesn't hurt the government, though. The first order effect is new tax revenues they don't have to source elsewhere.
As you pointed out, demand will drop, so there's fewer sales, and it drives inflation, slowing the economy, leading to (ultimately) a reduction in revenue all around. Also, it's basically a heavy sales tax, since it's through purchasing that's it's implemented, and sales taxes are always regressive (the poor spend all their money on consumables, the rich spend only a portion, so the poor pay more proportionate to their income).
There's a reason no economist these days is in favour of tariffs.
Only if the market will bear such expense. If your product sells at $100 and the tariff is $15, what makes you think the importer will be able to sell the same number of units at new price of $115? Depending on the demand, the $15 cost will be born by a combo of the importer and consumer.