Maybe I am still a little confused, but I travel 4 days a week, every week. I live in Hilton and Hyatt hotels across the US 180-200 days a week. I have never had an issue.
1) In the US, your stuff is safe in the hotel room. If you are nervous, hide it and put the do not disturb sign on the door and the maid wont come in. As suggested, stuff the room key and a $20 in your running pocket. I never bother with an ID.
2) Public transit....smell bad, who cares. You will never see any of those people ever again. I have been plenty smelly and gross in running shorts and a tee on the 6 train in NYC after a hard interval session on the LES track. If anything, I got a little bit more personal space.
3) Call, email, or check the website of a local running store. They usually have group runs. I actually met my fiance a year ago when I started working on Ohio by finding a local running store with racing team. My second week traveling to Ohio and I showed up to the weekly Monday night tempo run listed on their website. They had a place to change and leave my stuff. Even if their arent group runs, someone will be nice and show you a route or help you out.
4) I always have a rental car (expect in NYC), so I go on google maps and look at the satellite view to find a highschool track for speed work. Also look for rivers and parks, as there is usually a running trail nearby. Change in the car before your run and lock your stuff in the trunk.
5) You mentioned shopping? OK, odd but do it after your run. Or if you are near your office, leave the stuff at work till the next day.
All my traveling has been in the US except for 2 weeks in Mexico city. There I ran on the treadmill in the hotel gym. One day I went outside to run and after 30mins of circling the block because the bell man said that was the only "safe place for a skinny white American in those shorts" I got bored and went back to the gym.