I recently bought an older road bike ('88 Miyata 712) with very low miles and great condition. Frame size is 56 cm center to top.
I've been riding it now for approximately 150 miles.

Today I learned that I had took my body measurements wrong prior to buying this bike.
I am 5"11 and my inner leg is 2.82 feet ( 181 cm and 86 cm respectively). My long legs in relation to my height seems to be the problem.
My arms are also longer than average for my height.

According to most web resources, I seem to have bought a too small framed bike for my use. A suggested size (classic road bike frame style) is usually a 58 or a 59 cm.

I got the seat height right for me by fitting a longer seat post, but I'm wondering if this bike is going to cause me trouble as it is.
The bike sure looks funny with 15cm (almost 6 inches) of seatpost showing.

Do you think the size will be a problem for me on the long run? My goal is to ride at least a hundred miles a week.

As it turns out, by using the web calculators, the bike actually looks almost ideal size for my wife...
But If I end up hunting for a new, bigger frame for myself, how do I know I get the right size? I will be shopping in the used market for sure.

How did you learn that? You may have to go get a proper fitting. But IME, it is always more important to get the top tube length right--and with longer legs, shorter torso, and longer arms, you might be just right at 56... you didn't mention how comfortable the reach is, whether you feel cramped, stem length, etc. With an '88, it would be a little tougher to adjust stem length (although various-length 1" quill stems are still out there, just not as plentiful as the newer threadless design). The amount of seat post showing should be inconsequential. Just my .02

I measured my inseam length again, and noticed I missed by almost 3 cm the first time (the ruler wasn't straight, what a genius I am...)

Actually I haven't had a change to test for reach in a real way yet, since I'm still riding with a city handlebar. I have some drops and a 100mm quill stem on the shelf, but waiting for some brifters to arrive before I get to test them. I'm just worried because the bike looks so wrong with the seatpost sky-high. Also, the stem I've got is a pretty low one, how should that fair with my kind of body proportions?

Really depends on your flexibility OP. Your bars will be lower than your saddle and will be up to you as to whether this works out. Riders don't know what fit works best until they experiment. Some starting out like to have a professional fit to establish a baseline. If the bike doesn't work out and your wife is on the tall side, let her ride it and get a taller bike that will get the bars higher.

+1.

And six inches of seatpost showing is definitely not abnormal nor funny.

I have 6.5 inches showing for the seat post and have been professionally fit. Everyone is different.

If what they sold you is a 58 or 59 cm bike, then you're pretty much on the money. A larger frame would likely mean reaching too far forward for the handlebar, and smaller would mean reaching too low. One advantage to having long arms is being able to reach lower without upsetting things too much.

As a data point, I'm 182 or 183 cm tall, with 89.5 cm legs. I've been riding bikes this size since 1970. Eddy Merckx is 184 cm (or was before he got old), with long arms and legs. That's his size, too.

Yesterday I tried fiddling with the riding position again, and I noticed that I could not move the saddle far enough back for me to reach KOPS - the nearest I got was about 1 - 1.5 cm forward of KOPS.

How much of a problem is this, and what does it tell about the frame sizing and/or geometry?

OOPS - KOPS - bad juju for the 41... shouldn't have brough that up

but anyway, KOPS, measure from the Tibial Tuborosity not the front of the kneecap... Tibial Tuborosity is about 1-1.5 cm behind the knee cap - so, fuggetabouit...

56 cm Older Miyata - likely sortta normal ST angle, so it should workout OK. a decent setback seatpost should handle getting the saddle in a good spot.
your biggest issue might be getting a long enough quill stem - 120ish likely, maybe 130ish...
if you don;t find one, then there are adapters so that you can use a modern stem ...
ride it for a while, get some miles, get some experience - the bike buying FLU will hit soon enough...

Don't worry about KOPS for now. Get your saddle to the proper height if you have done the fit yourself it's probably low. 109 x your inseam, pedal to the top of your saddle is good place to start.

Now this is a noobish question (but I am a noob in road biking...) Is inseam and inner leg length a different thing? I have measured the distance between the pubic bone and ground straight up with me standing normally, would inseam have to be measured along the leg? I used the "ankle just scraping the pedal at its lowest position" method, and the height seems about right.

How will i know when the saddle it too high? Is it only the 'hips not rocking while pedaling' that tells the saddle isnt't too high?