Linguistic Term for “Shame if something happened to it” type phrases

They could be one of two things, Gerunds or Deverbal Nouns.

The wikipedia article on Gerunds has a decent section on the nominal and verbal properties of gerunds.

Nominal characteristics

1. The gerund can perform the function of subject, object and predicative
2. The gerund can be preceded by a preposition
3. Like a noun the gerund can be modified by a noun in the possessive case, a possessive adjective, or an adjective

Verbal characteristics

1. The gerund of transitive verbs can take a direct object
2. The gerund can be modified by an adverb
3. The gerund has the distinctions of aspect and voice

EDIT

Based on feedback I'll point out a specific section of that Wikipedia article.

Not all nouns that are identical in form to the present participle are gerunds. The formal distinction is that a gerund is a verbal noun – a noun derived from a verb that retains verb characteristics, that functions simultaneously as a noun and a verb, while other nouns in the form of the present participle (ending in -ing) are deverbal nouns, which function as common nouns, not as verbs at all.

So from your examples, the grammatical term you might be looking for is Deverbal Nouns. Only a subset of deverbal nouns end in -ing, however.