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In German this would be the verb zurechtkommen, it's meaning being very close to to succeed, but emphasis being more on the not failing part.
Examples in English:
In the first of your three example sentences, I would use elturniĝi (this is also the only sentence in which I would use zurechtkommen in German). In the other two I would use sukcesi.
Generally, I would say sukcesis, but if in certain contexts you're going for a different nuance, there are other potential options.
Benson (CEED) and Wells suggest alternatives such as helpi al si. In some of the contexts you gave, both sukcesi and povi trakti seem to work.
Your final example suggests to me possibilities like mi ne povas tajpi... or *mi iel neniam sukcesas tajpi...".
If sukcesi seems too easy, you could always clarify with additional words:
I struggle to find a universally applicable translation for to cope, to manage (intr.) Maybe atingi siajn bazajn celojn, or tra(ns)en, tra(ns)- combined with a strenuous verb (peni, strebi, etc). Your own negative version is possible: ne malvenki, ne malsukcesi, eviti malvenkon.
Estis malfacile eduki du infanojn sola, sed ŝi...
Vi devos inversigi la polusecon de la neŭtronfluo.
Helpu! Mi devas sendi mesaĝon sed...
I haven't searched for one, but off the top of my head, the point of coping/managing is that you're contrasting it with there being literally no way to do something, yes? So further down the scale of confidence from a sukcesi/malsukcesi contrast I posit would be an iel/neniel contrast. Slap on the ever-useful -um- suffix and I would use something like ielumi.
Example:
Mi ne scias la kielon, sed mi certe ielumos.
I don't know how, but I'm sure I'll manage.
