1. Notice to be extended to project ending time. He threw in resignation on 2nd Sept and was expected to join new company by early December but his project will end the week before Christmas.

make sure you have evidence that you submitted the resignation on 2nd Sept, eg. email or etc

2. No leave can be used to offset his notice period. These leaves will be considered burned as no cash is payable to him.

this is illegal i suppose? you are entitle to the leave prorata to the year you work, either they pay you in money or let you go earlier

3. If he chooses to leave early, the company can sue him based on breach of terms of the contract.

maximum 3 months as stated in the initial offer letter you signed
just say "NO" to anything extra, they cannot force you to work
If they play ugly to force you, it is 2 ways street actually, since you're leaving already, there is nothing to worry about performance, bonus and etc, you can play the game too, by doing the minimum you have to do, finding legit and valid excuse why thing gets delay and need extra time.
they will soon realise playing dirty and upsetting you wont bring them any good nor extra manpower too

Problem is his initial offer letter did not have this and he was not asked to sign any additional suffix documents that indicated the change in the notice periods.

What should he do?

Bonus Question: The company even rejected the new company's offer of buying him out saying what they need is not money but a person to run through the whole project. Is that even legal to reject a buyout?

yes, the company can reject the offer
but again, it is 2 ways street as above. while I dont recommend tear the face or burn the bridge, but you just have to play the game cleverly
it is just a game of negotiation