Runners - What do you pack?

I just weighed the contents of my bag after my last trip and my running gear is both bulky and heavy. I have over 5 pounds of gear for running in my bag.

This was a 4 day trip in which I ran 3 of the days.
I brought 3 sets of running clothes (underwear, shorts, synthetic shirts, 2 cotton socks, 1 smartwool PHD) 1 cotton long sleeve tshirt that I wear on top of my synthetic shirt. Temperature was in the low to mid 40s. I keep this all in an eagle creek clean/dirty cube.

I have not optimized my running gear for travelling at all, but I am looking to start. This trip (and some others) did have a situation where I ran in the morning, had to checkout right after, do meetings all day, then check into another hotel in another city. Which makes me think I need two sets of clothes for those trips. However, for other trips I would like to get that down to one set.

For most trips where I don't need to hotel hop, I want to get this down to one set of clothes that I can wash in the shower with me and then hang up to dry before the next day.

• Socks - The PHD smartwool are OK. I am considering replacing with darn tough. Its hard to decipher which of their socks is best for running. Any opinions?
• Shorts - I have some nike that don't have a lining. Recommendations for light weight, packable shorts with a lining? I think if I have a lining I can ditch the underwear.
• Shirts - Nike Dri-Fit is what I have here. But they are pretty heavy. Any recommendations for more packable and hopefully quicker drying shirt? Sometimes, depending on the weather the heavy might be better anyway.
• Long Sleave Shirt - This is where I need the most help. The cotton shirt is stinky and takes forever to dry. I usually like to layer on my cold runs. In general, however if the weather is below 40 I am just going to run on the treadmill so I don't need something super warm. Is ice breaker my best bet here? If so, any suggestions on which ones?