請問動名詞使用

(1) 請問following the news 是否係 which follow 的簡化寫法

No, strictly speaking. "following" here is a preposition (meaning: [在...以後] after an event or as a result of it ).
more examples: (Longman dictionary)
Following the president's speech, there will be a few minutes for questions.Thousands of refugees left the country following the outbreak of civil war.(2) resolving 可否改為 to resolve

No, it cannot. "resolving" is a gerrund, adding "the region's debt crisis" to form a noun phrase which is the object of the preposition "toward".

(3) I have the right to know that who is my partner in the party.
to know 改為knowing 得吾得？

No, unless you put an "of" before "knowing". ~ right of knowing

Common uses (according to dictionary)
- right of: e.g. right of abode, right of appeal
- right to: e.g. Everyone should have the right to freedom of expression.
- right to do something: You have the right to remain silent.

Received an inquiry for the use of gerund as underlined for the briefly ans-----------------
"following"participle to "stocks surged about 3%"
"resolving-------" gerund to "a step forward'
to know precedes (to know who)