#2: Definitely bring a U-lock unless you are sure you will be seated where you could see your bike at the restaurants. But most U-lock would take away a bottle cage space or significant amount of luggage space. I personally use TiGr Bow Lock - the standard or long one could lock both wheels and the frame to a pole, and could be strapped along the top tube and save precious luggage space:
https://www.tigrlock.com/tigr-bow-lock/
#3: I recommend following the river trail until you almost reach the Lincoln Memorial. Use the crosswalks to get to the sidewalks of Constitution Ave, head east until you reach 15th St. Turn left onto 15th St, and at the third block, turn right onto F St. The F St is relatively quiet and has sharrow lane. Also, it goes straight to the Union Station. If you plan on using the bike roll-on service, make sure you arrive an hour before the departure time. It generally takes about 30-40 minutes to get there.
P.S. I rode straight from Harpers Ferry to DC to catch the train for the past two trips. The reason was that I found accommodation in DC too expensive, and the Harpers Ferry Hostel is comfortable and cheap ($25 per night including breakfast). I left at 7 am, and arrived at mile 0 marker around 1 pm - just enough time to take a few pictures and grab a sandwich for lunch before heading toward the train station.