You're trying to convert the wrong part of the phrase. The passive bit is "have been forgotten". An active rendering of this would be:
Things that we should not have forgotten.
or if you want to continue using "no-one":
Things that no-one should have forgotten
but I don't really think that preserves the original sense.
Told many times, he finally understood it - Doesn't sound right.
I would say "After being told many times, he finally understood it."
The first action should be completed - first he was told, then he understood. In this case, it is the same as 'having been told'.
Being told the truth, he looked shocked - two actions are simultaneous. When he was ...
'That she be placed' uses the subjunctive mood. We can use the subjunctive in two main instances: wishes or hypotheses, and imperatives. For an imperative (a necessity, a suggestion, or an imperative action), we form the expression using 'that' and the bare infinitive of a verb:
Necessity: it is essential that Peter send the money today.
Suggestion: he ...
This is an example of one of the meanings of 'only':
You can use only before an infinitive to introduce an event which
happens immediately after one you have just mentioned, and which is
rather surprising or unfortunate.
Speedy Gonzalez was banned, only to be returned after mass protest.
My brother won $100 on a bet at the racecourse, only to lose ...