Here's what I would like to know about the officers in the conventionally powered Navy:
1. How are officers selected for engine room duty?
During your 2nd tour you can qualify Engineering Officer of the Watch (EOOW) and get a letter. This is mostly voluntary on the officer's part, no one is going to 'select' you to qualify EOOW. Past that, you can be detailed as the Chief Engineer (CHENG) as a DH, which is typically a 2nd tour DH job but not always. You don't necessarily need to have an EOOW letter to be detailed as a CHENG. Also, the CHENG doesn't stand 'engineroom duty.' He is in charge of the material condition and readiness of the entire ship as well as the operation and maintenance of its propulsion machinery. His normal watchstation as a DH would be either Officer of the Deck (OOD) or Tactical Action Officer (TAO).
2. Is there formal engine room schooling?
Not for conventional SWOs.
3. How are they mentored on the job?
By constantly being yelled at for every mistake until they finally deduce the right answer by all the ways they've been yelled at for the wrong answer. I'm not kidding. Also, by being yelled at for sleeping or anything else that doesn't involve work.
4. Any secondary duties?
Yes. As a division officer you'll have collateral duties depending on what the ship needs at the time you report.
5. What, typically, is the highest level of command duty for a junior officer?
OOD underway. The EOOW is subordinate to the OOD, although apparently they don't really 'talk' on a conventional ship.
6. Are intermediate staff positions available outside of engine?
No.
7. Can a chief engineering officer become a ship's captain?
Yes. In fact, CHENG is considered a tough job and if you do well you will generally get competitive fitness reports for command.
If so, how many years service does it take to get command of a destroyer or frigate?
Typically around 16-17 years. FYI, frigates no longer exist, you mean LCS.
8. Can an a mid career engine officer be assigned as a base or port facilities manager?
No.
9. Can engine room officers go on to schools like the Naval Warfare College or achieve flag rank?
There are no such thing as 'engine room officers,' you are a SWO and the Navy codes you as an 1110. You promote (or not) just like every other 1110 and every other unrestricted line officer (sub, pilot, nfo, seal).
If you mean engineering duty officer, that is an entirely different track and would require a tour on a ship to get your SWO pin. After that you go into depot maintenance management or system engineering tracks. I'm not an EDO so I'm fuzzy on the details on what that entails, but it does involve a stop at either Naval Postgraduate School or MIT to get a master's in system's engineering. It also requires that your undergrad be in engineering.