This is a complicated question because you're not specifying which "doing" you mean.
The word "doing" can be several different things.
1) Noun. This is the only case in which a singular and a plural exist.
Example: "Is that mess in your bedroom your doing?"
"The shady doings of my husband caused me to file for divorce."
As a noun "doing" or "doings" always mean an action.
2) Verb. Doing can be two different forms of the verb "to do."
A) Present Participle.
In this case it expresses continued activity. Example "I am doing the best I can."
Gerunds are used the same way as nouns. Example: Doing the right thing is
I am not sure what you mean by "activity without a goal" as the word "activity" implies some kind of action by definition.
If you were to use the phrase "doing nothing" there's no specific goal involved but some
people would say "nothing" is the goal. In other words, to be inactive. I can't think of a case where you can use the word "doing" whether as a noun or a verb to show actions without intended results.
When a project is shut down it is said to be no longer active or to be inactive. Maybe that's what you're looking for. No "doing" is used here.
Procrastination is doing nothing because you're unsure of how or when to do it. The verb form is to procrastinate. Maybe that could help.
The use of "no doing" means lack of action. For example "I was seen as irresponsible through no doing of my own." It means I didn't do anything to deserve to be seen in that light. Again no activity.
The word project doesn't have an opposite. It's just be the lack or nonexistence of a project.