1.He delayed doing/to do his homework.
2.He suggested going/to go on a picnic.
3.He quit smoking/to smoke.
4.He avoided meeting/to meet the lady.
5.He proposed going abroad/to go abroad.
(Which one is correct? Do we have to use only gerunds in the sentences above?)
5. going abroad
Yes, gerunds -- they are serving as direct objects in these sentences.
Yes. Only the gerunds fit in those sentences.