To answer the title question: I don't casually pass as female, and I don't put any effort into passing because I don't really care, and I don't correct people when they misgender me. So in that sense I guess I represent as male. I'm still getting rid of my penis cause I need to (pretty much for the same reasons as you), it's happening real soon. So yes.
I've seen and heard of people who want to transition to change their body, without wanting to transition "socially". That's not weird. There isn't a path to follow as many doctors and some trans want you to believe. Transition consists of a whole bunch of things you can chose to do or not. It's up to you.
I wasn't in a relationship when I began my transition so I can't really give you an advice with regards to yours. If your wife knows that you're trans it might help.
HRT doesn't make it impossible to get/maintain an erection, though things work differently. From my own experience, it's a matter of whether you remain sexually active or not.
I could be wrong, but I think those who experience loss of erection and libido or atrophy are those who stop having any sexual activity during their transition. It's pretty common, cause we don't like our penises.
HRT is said to make your penis shrink but oddly enough I had the opposite experience, it keeps getting more and more difficult to hide when I wear tight clothes.
I've been on hormones and anti-androgens (sometimes at high dosage) for 1 year and 6 months, and I don't have any libido or erection problem. Ejaculations are nearly gone though.
But remember that the effects of HRT vary greatly from person to person, nobody can tell you what it will do to you and whether things will work the same way as they did for me.
My advice would be to start going through the steps to access HRT and SRS. I don't know how much time it takes where you are, but here getting access to HRT can take months, and getting access to SRS never takes less than 3 years.
So I suggest you start doing all the paperwork and medical stuff that it takes, just so it's already done if you decide to go through medical transition.