Ne is indeed mostly skipped in speech, almost universally. Some probably take pride in using it systematically, and I know of at least one person who had French as a second language and was unable to let go and follow the common usage, perhaps not noticing how unique her speech was in daily life. Her kids more closely followed the regular usage, despite the fact they were all living in an English-speaking town far from any French-speaking community, and were therefore not using French as much as they might have been elsewhere.

There are however, as you point out, exceptions to the disappearance of ne even in day-to-day regular speech. Firstly, though this is not what you’re after, in complex structures where two negations meet and would otherwise force the encounter of two consecutive pas :

• On pouvait pas ne pas remarquer sa gêne.

And secondly in some short, almost fixed sentences, sometimes used as a sort of jest, but also occasionally without any intent of being funny :

• Je ne saurais dire
  On ne saurait dire pourquoi / comment
  Je ne sais que dire
• On ne peut le nier
• Je ne puis l’accepter / le tolérer
• and obviously the world famous je-ne-sais-quoi
• …and the ubiquitous n’importe in all its variations:
  → n’importe comment, n’importe où, n’importe quand, n’importe quoi, …
  including the substantive n’importe-quoi, as in “C’est vraiment du n’importe-quoi, cette réponse !”.

Versions containing pas, skipping or not ne, also exist for some of these examples, but I have never encountered these :

• To avoid (as far as I know): Je (ne) sais pas que dire
  which could however be changed to (the familiar): Je sais pas quoi dire
      (though “on saurait pas dire”, strangely enough, is common around the area where I live)
• Best avoided (I believe): Je (ne) puis pas l’accepter
  → (but “Je peux pas l’accepter” is common)

I can’t help but notice that these last two examples use uncommon features. The first one an acceptation of que close to quoi, that one may encounter in questions before a verb (Que dire de ceci ? Que conclure ?), the second a rarer form of conjugation of pouvoir (je puis). Could these features force upon the speaker the use of ne without pas in negations? I wouldn’t say no.

Concerning your examples, and their potential usage in emails, here is what I would consider if I were the one using them. They are strictly personal opinions, and you may want more than one advice, but it could give you some guidelines.

• On ne peut le contrôler
  → The relative pronoun le makes it a bit complex, and one may miss ne in a longer text. However, “On ne peut tout contrôler” would not pose any problem to me.
• Je ne puis l’accepter.
  → Perfectly fine. No problem.
• Des idées qu’il n’osait exprimer.
  → Probably fine even in a larger context.
• Je ne sais comment cela se fait.
  → Perhaps a bit precious, though I wouldn’t wince at “Je ne saurais deviner comment faire / comment on s’y est pris pour le faire”.
• Je ne sais pourquoi cela est arrivé.
  → Once again, on the precious side. A simple “Je ne sais pourquoi”, obviously in a different context, would be okay.
• Je ne sais que vous dire.
  → This one is also a bit complex, because of vous. Other similar sentences would be fine “Je ne sais que dire” or “Je ne puis vous répondre pour l’instant”, or even “Je ne saurais que lui dire”, but this one in particular, addressing the comment directly to the Reader, is strange to me.
• Les choses n’ont cessé d’empirer.
  → No problem with this one.
• Si ce n’était trop tard, je le lui dirais.
  → I believe this one is fine, though I would probably tend to prefer “S’il n’était pas déjà trop tard, je le lui dirais”, which uses pas.