Which Branch Should I Join as a Civil Engineering graduate (Army, Navy or AF)?

Definitely Army.

I'm not going to answer your question, I'm going to give you a non military option. US Public Health Service. They are one of the Uniform Services, all members are commissioned, equal to a military commission. Your uniform is basically a Naval uniform, with a slightly different emblem. You get all the same benefits as the military, and can transfer to the military at a later date.

If you want to travel to build things it's a toss between the AF or NAVY. The USAF has travelling CE squadrons called RED HORSE. They build anything and everything you can imagine. NAVY has CBs, construction battalions. They do the same thing but been at it since WW2 It's your choice really, choose one, but choose wisely.

I was a 2d generation Navy Seabee so I'm partial. I'd recommend the Seabees for a number of reasons. The "Bees" are the top combat proven builders the military has. As a reserve (depending on the unit) you will be able to do 'public service work' in your area/region. As military "tradition" goes, the armed forces will tell any other that they are the best, but the main units that get little to no "flack" are the "Bees". Past history dictates that it you wanted anything fixed the "bees" could get it done, and nobody gets in the way of a unit that can get "fixing" done whether it's a truck, building, tower, road or whatever is needed.

1) The jobs that engineer officer in USACE does is not real engineering but more like management.
-- Most Officer Jobs are management, it doesn't mean you won't learn, but you focus on managing different levels of units. A 2nd LT of course manages (leads) a platoon, Captain, a Company, and up and up.
-- As an Officer you also will work in sections, S1, S3, S4, etc. and focus on planning. Taking Operations Orders analyzing them and coming up with the best recommendations for your commander and the unit. Yes, people will say staff turd but trust me you learn things working with operations orders and planning.
-- I will also say this too that even as an NCO, the higher up you go, in the NCO core, you do the same things focusing on "a" specific area, and many times as an NCO you have more experience than the officer in that particular area, thus advising him/her. I have to mention this as well. You learn to rely upon your NCO's.
-- Even as an NCO the higher up you go your managing (leading). A First Sergeant may say babysitting.....LOL. Just because you have a degree doesn't mean you have to be an officer. I have a friend who is an SFC who is also a Vice President of a startup company. He has his MBA.
-- I guess a good example I can give is a Signal Officer, signal officers are more managers while the enlisted know the details of the equipment. The Signal Officer focuses on planning and management. But the signal officer, if they take the time, will learn the systems, and details, it makes you a better leader to your soldiers.

2) My strong interest is structural/design analysis of either buildings/bridges/dams/ships.
-- I can understand the need to find something related. I am infantry and am a software engineer. Try to find something that matches to the best of your ability. I chose infantry but will be moving to Cyber.

My information is going to be a little bit dated. However, I did work in the engineers for the Air Force. It was my last gig with them. There were officers and warrant officers assigned at the Squadron level but they are more likely I believe to be found in a logistics session. They did work on projects they had to do with the PE side of Base planning you might want to investigate that. You could be a warrant or regular officer either way.

There isn’t much engineering in federal service. That is all contracted out. You might want to stay in the Civilian sector. My daughter is an electrical engineer. She is actually building a robot this month. The Army is making her a field artillery officer. Just because you have an engineering degree doesn’t mean you will get to branch engineer. Unless your a doctor, nurse, or lawyer the Army doesn’t care what your degree is in.

Marine Corps, Grunt ! You will never forget the bond of this brotherhood and you have a lifetime to get your career in order and a life time as. Marine to cherish with life long friends and brothers you will never forget.

I can t comment on which branch might be better, but I can point out that age matters if you are going for a direct commission. I took a mid-career MBA, in night school, and so was close to the age limit to apply for a direct commission when I graduated. That s when I found out that there were some additional pre-requisite courses the Army wanted done, and by the point I was exhausted from the grind, and just dropped the application, as completing the courses would have pushed me over the age limit. In hindsight, I probably should have looked into a waiver, as leadership courses I already had on my record would have probably counted. 20-20 hindsight.

The choice is up to you however in the Army you need to look into the 21 series that the MOS for Combat Heavy Engineers. In the series you will find carpenters, masons, electricians, and plumbers just like you find in the Seabees (Navy), or Redhorse (Air Force). But, you must understand that during wartime we really don't a lot of reconstruction projects. During peacetime we will be doing projects that fall in the area of civil engineering. The Army has constructed roads, school, housing, airfields, wells and much more. But, like I said the finial choice is yours talk to all of the recruiters before you jump into anyone branch.