You're better off discussing with your employer.  Some won't care as long as you're not going after competing work or you let it affect your ability to get your work done at the office.  Others have very strict policies--mostly due to overlapping risk and liability.  

Be transparent and let them know your desire to work on other projects.  Perhaps they will see it as an opportunity for growth--both in you as a project manager and for the firm (ie. new work or typology).  Your new role would undoubtedly lead to more money.  

I think I addressed all three of your concerns.

There might be etiquette... but fuck that.  The real issue is liability with your parent employer.  If shit goes sideways, it may be difficult to prove that your employer's firm did not have a part in the project.

Our office just recently changed it's internal policy effectively banning any moonlighting after it was discovered someone was using office equipment for personal client works (also, the individual was unlicensed).  Prior to that, the office was rather relaxed about it.

Side work must be done entirely on your own time and entirely on your own equipment and software.   To avoid the appearance of competing with your main employer, it is also wise to do side projects that are not in the same market sector or size that your day job does.

Some firm owners may welcome employees doing side work and even send them projects that are too small for the firm.  Others are very threatened by it.

In my case, side work was an invaluable gateway to owning my own firm.

I came in one Monday and found a drawing lying under the plotter, for a
project that wasn't ours.  I looked at the key card log to see who had
been in on Sunday and had a talk with that employee.  Our policy in the
employee manual is not a blanket no-moonlighting-at-all policy - but it
prohibits using office time or resources, and it prohibits conflicts of
interest.  Our employee could not claim with a straight face that she
wasn't using the firm's resources, since she was plotting in the
office.  But I realized in talking with her that she didn't fully
understand (or pretended not to understand) that things like sending or receiving emails on our computers - even if not using her work email address - were a problem.  Or that responding to emails, texts, or calls on her own phone but during office work hours that she was billing to our clients was a problem.   I walked her through all the things I could think of, related to outside projects, that shouldn't be happening in our office.  I explained several scenarios that could be conflicts of interest.  I explained that, theoretically, we can lose our insurance and/or be exposed to liability for her projects.  I suggested that she consider trying to get her clients to hire our firm, if she felt they needed the firm's resources.  She seemed to absorb all of that information.

But... despite that discussion, a few months later the same thing happened again, so we let her know it was time to part ways. It was as much because of the sneakiness and disregard for what I'd said, as it was for all the liability/insurance reasons. I would say the "unspoken rules" are: don't use the firm's technology or supplies to do anything related to your outside projects; don't use time that you're recording as working for the firm to do anything related to your outside projects - do them entirely on your own time; be honest with your employer, get their blessing, and fully understand from them what could be a problem; and if you're going to be dishonest anyway then be very careful about cleaning up the evidence!

"(like a new $35 million highschool)"

That doesn't sound like a side-gig territory. Are you just a junior drafter pinching in with production? $35 mil projects come with some very serious deadlines that require undivided attention for weeks.

I get offered side projects all the time, and I always reject. Smaller projects only require me for about 20 hours of involvement, but it is not possible to keep this work contained to evenings and weekends. It will either be conflict of interest with my other commitments or an inferior service. 

But, architects doing residential on the side where they landed their own client and where client prefers off hours communication anyways, and where all work is done in off hours, is fair game. I have seen people do that and it has not interfered with their main gig. 

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Do it, but create a legit separate company so that your employer doesn't have any legal liability  / exposure.  Particularly if you aren't competing with them for work ( that's awkward and just inappropriate )

Avoid any correspondence or communication with those moonlighting client(s) during your other gig.  It's how we all started out, just don't be a dick about it. Be professional from day one.  Then when the right "big fish" client comes to you for work you can decide if you are ready to leave the comfort of steady gig or open up your own practice.

We do not allow our staff to moonlight.  It's a big deal.

It's a question of liability for the firm.  If they screw something up on the moonlight job, the first thing the client's attorney will do is look for deep pockets with insurance.  That's most likely us.  It's nearly impossible to separate the moonlight job from the firm in those cases.  Did the staff member either receive or send any email about the moonlight job from their office email account?  Did they take any phone calls from the client or the builder at the office?  Did any drawings get delivered to the office?  There are all sorts of ways that an attorney can connect the job to the firm, and once they do, bingo!  They can go after the firm and the firm's E&O insurance.  There is established legal precedent for this.

It's in our employee manual, and we let every new employee know about the no-moonlighting policy before they agree to join us.

Only with consent from employer - and advisable to have own insurance.  I have always taken issue with those that moonlight even on their own time and equipment as inevitably it eats into productivity and the burden of covering the work in the office weighs on others.

I have in the past done pro-bono work with permission but more presentation and planning level drawings that were not for construction.  The limited liability was worth the exposure the firm got by having employees do the work for non-profits.

Reasons for moonlighting having been forbidden by employers: liability (whether licensed or not, the employee has at the very least training and knowledge that is being used to do the work), cost (if using equipment, or if the moonlighting is eating into productivity), morale (if individuals moonlight and others carry the work), competition with firm.  Work gained for moonlighting is often in part do to the reputation of the firms work. 

I find it sad how often people who moonlight using office equipment are unaware that activity on the office equipment is easily tracked in numerous ways.  Even seen a few blatantly do it during the day or while logging in from home.

On the note point of keeping things fresh, I've seen firms where they encourage small competition entries or in-house design charettes to get everyone in the office sharp on design and representation/graphics.  I get that this is a bit of overhead cost, but seems like a worthwhile thing for offices that have a bit of financial security.

No moonlighting because it hurts the firm but when layoffs come around they say it's "just business, work on your own projects to stay fresh"

1st RULE: You do not talk about [moonlighting].
2nd RULE: You DO NOT talk about [moonlighting].