But what about sentence 2?
2. What we discussed about our new project yesterday needs to be further developed.
Is this sentence grammatical without "about"?
If not, why do you need "about" in sentence 2 ?
You need "about" in sentence (2) because you have a discussion about something. In sentence (2), the interrogative pronoun "what" stands for that "something."
2. What (i.e. The point (that)) we discussed about our new project yesterday needs to be further developed.
Hi, Apple and Gustavo—I think it is possible to parse the "about"-prepositional phrase as modifying "what" in the structure underlying (2), and as modifying "point" in the structure underlying your paraphrase, Gustavo:
the point that we discussed about our new project yesterday
= the point about our new project that we discussed yesterday
what we discussed about our new project yesterday
= what about our new project we discussed yesterday
Consider the following dialogue as an illustration of this. In it, we can see the "about"-prepositional phrase modifying "anything." Essentially, then, this is the same grammatical structure (complementation pattern) that we have in (1):
A: What we discussed about our new project yesterday needs to be further developed.
B: I beg your pardon. Yesterday, we didn't discuss anything about our new project.