Asher Black wrote:

and the single quote key and ^ key on the right don't advance the carriage.

This key is suppose to function that way. Is a "dead key" and is used for making diacritical marks (accent marks) above letters.
 

It's a Kolibri! Never write it off as worthless, if you have a Kolibri that types like a speed demon you have treasure.

I wonder if you just need to give the workings a good clean with eg cotton buds and solvent... the fact that the backspace also plays up sometimes makes me thing there;s a general stiffness/dirt issue in your innards. Trace the action when you press the keys and clean all along the levers etc... sorry if I'm teaching you to suck eggs! It looks from your picture of the type bars asa if the segment could do with a clean too. In my experience, cleaning - and then cleaning some more, and some more - can work miracles. 

Re the ' key - do you mean the 8? - not advancing the carriage, is it maybe missing its little spring? Or maybe you need to get a cotton bud (gently, being careful of springs) into the wheel of the escapement... 

I have a Kolibri that's missing one of its carriage knobs, and the top of its backspace key, but I forgive it because it's such a great typer, and so very beautiful and tactile.
 

OK. I'll do as good a cleaning as I can do without taking it apart. THere are no screws on the bottom plate to take it off - just the screws that hold on the feet. So I don't know how to get in very far. I'll do my best. Thanks But are you saying both the ' key and the ^ key are dead keys? Those are the 2 with the arms different, so it correlates. And any notion what causes the margin issue?

Left margin adjustment is pretty common fix on the machines I've gotten. I don't know anything about your Groma Kolibri, but on the American machines there is usually an adjusting screw on the left margin set on the carriage. (But, on my old Underwood desktop it took the removal of a couple of thin washers on the right side of the margin rail under the index bar.) sometimes you have to remove the paper tray or even the tab rack to get to the adjustment screw.

So I think all issues boil down to a single thing. Nothing is dirty. Nothing needs oiled. All springs are there, and parts move freely. It's clean and pristine. But apparently you can take off the bottom plate. And what I found is:

I know: basic typewriter stuff: I don't know what you call all of this but: There's a pulley with a wire that advances the carriage, and there's a toothed wheel that catches a toothed bar, which corresponds to individual spaces/places on the typewriter. When you type, hit the space bar, or backspace, it moves one of those spaces. The pull with wire pulls the carriage along, which has the fixed toothed bar, and it catches in the toothed wheel. All good.

What's happening: in some cases, the pulley/carriage does not move that one space/place. It works great on all typing. Where it fails is on backspacing (sometimes) and on the beginning space on a margin on the carriage return. You can see the the pulley just doesn't advance all the way sometimes. Whether that's the wire, pulley, toothed wheel, toothed bar, or something else, I don't know. But it's not a left margin setting or anything to do w. the backspace linkage.

I've looked at every tooth on the wheel and every tooth on the bar, and they're all fine. The wire isn't frayed - it looks fine. The pulley is tightly screwed down with a tiny screw. And nothing seems amiss. I don't know what this could be.

The toothed wheel is to the right and just above the pulley in this photo (covered by something w. 3 small screws).