“feared to lose” doesn’t sound right to me. When you “fear to do something”, it usually implies that you have some control over whether you do it or not. For example, “I feared to ask him”. Losing something, however, is something that happens to one involuntarily.
However, there are enough plausible-looking Google hits for “feared to lose” in a relevant context to make me believe that some speakers may accept it.
There are, as you know, certain verbs that are followed by either an infinitive or a gerund. In some case the meaning changes according to which one you choose. Verbs like ‘try’ and ‘remember’ are examples. For me the use of the infinitive tends to point to a specific activity and the gerund to a general one for verbs that can take either, In your example above both constructions are fine. The only difference for me is that the infinitive ‘to lose’ indicates a particular incident and the gerund suggests something more general.
“I remember to post your letter” sounds strange to me. For future action I would say “I’ll remember to post your letter”.
I do not fully agree with this distinction. “I tried to tell him” and “I tried telling him” can both have the sense “I told him but he didn’t listen”. “I tried to tell him” can additionally have the sense of “I made an effort to tell him but was unsuccessful”.