Which bank performs the currency conversion during a cross-border wire transfer?

This is a treasury/FX function. From Sweden-to-US, the deal will most likely be executed on the Swedish side, where the Swedish Bank's FX desk with purchase Dollars being quote on the USD/SEK pair. This will most likely happen in Sweden, and then the Dollars are forwarded to the American Bank. The US bank would most likely not be doing the FX conversion.

What would the US bank do with the SEKs?

The only variable is that the FX dealer could be based in Sweden, or in Europe. Unless some FX dealer in the US would like to take SEKs.