For travel...either semi-formal or knocking-around-casual, my husband mostly wears the Brooks Brothers no-iron long sleeves shirts. They pack well and stay crisp looking for the entire trip.
The way we pack is to put all the hanging garments (this would include the jacket and trousers, too) in a very light-weight garment bag, then fold that into threes and put it in the RickSteves rollaboard. If we are doing a one-night stay often I will just unzip the bag and fluff items out, but leave them all on the hangers within the bag (hope that makes sense). But, for multi-night stays, we take all the hangers out of the bag and hang them like we would at home.
Brooks Brothers shirts at not inexpensive, but they typically have really good sales at least twice a year (which is when we stock up).
For ties, I told the tie in half and then roll it up in white tissue paper, then put the rolled up result into a round Tupperware type container (and old clean sour cream container would work just as well). We used to hang a tie over the trouser bar on a hanger, but the tie could get smooshed or wrinkled....so the roll--up trick, protected by something a bit firm tends to work well.
I would never try to sink wash a dress shirt or even a nicer casual one. The only men's shirts I would consider sink washing are the SPF protection type or a very casual short-sleeve shirt. The Brooks Brothers ones don't do well all balled up and then hung up, no matter how much stretching and smoothing one does. They might do okay if sink-washed and then tossed in a dryer on low for about 4-5 minutes, but wet balls in a sink and hung...I'm thinking not.
Have a fun trip