Not necessarily: the company can choose to absorb part or all the tariff, since the demand would drop at the higher price anyway, and they might make more overall profit at a lower margin per item. But generally yes, most of the cost will be passed on to the consumer and prices will increase on average.
The difference is that this way it's much easier to calculate prices.
If the tax were 20%, the exporter would have to do the inverse calculation. That is, "which price will result in me gaining $1000?" Which is not 1200, since 20% of 1200 is 240. x = 0.8y -> y = (1/0.8)*x -> y = 1.25x. so the exporter would have to price it at 1.25x the price, $1250. 20% of 1250 is 250.
So it's unintuitive that a 20% tax would result in a 25% price increase. That's my guess why tariffs are applied to the importer instead of exporter.
The only difference would be that money we spent would be going to the companies instead of the government. Tarrifs are a government putting taxes on their people to strangle industries in other countries. In both scenarios we pay the same, but the flow of money is different