When using bullets, I look at them as alternative endings to the introductory sentence. In this example, the introductory sentence says "The company's reputation increases when...", the bullet points then have to make sense as a continuation.
Bullet 1 becomes
(when) interest is Piqued/stimulated/captured at the government level gaining involvement from both government and non-government organizations.
Bullet 2 becomes
(when) interest is increased by the national and foreign media
(when) increased interest is shown by national and foreign media
Bullet 3 becomes
(when) decisions can be made without substantial work engagement.
(when) there exists a possibility of making decisions on reduced work engagement
(when) the possibility of decision making on reduced work engagement exists
So in answer to the original question, "possibility" can have either a definite or indefinite article depending on the context. If you previously mention the "possibility" or use it in the context that "the possibility exists", making "possibility" something specific then the definite article can be used. If you are casually referring to "a remote possibility" and have not previously mentioned this possibility, then the indefinite article can be used. Personally I would use neither and confidently state that "decisions can be made".
In answer to the 2nd part of the question, I would not use an article on interest because "interest" is not being used as a noun, interest is being used as a verb. To show interest, to be interested in (verb). Not "the public interest" (noun). I would also not use an article on national media because media could be considered countable (plural).