The lulav was waved during the hallel by hodu in the begging and in the end and by ana hashem hosheana, so says Bais Hillel. Bais Shamie adds even by ana hashem hatzlichana. Reb Akiva says I was watching Reb Gamlel and Reb Yhoshua and they only waved by ana hashem hosheana.

Somebody who was coming form the way, as soon as he gets home he should take the lulav. If you did not take te lulav in the morning you can take it all afternoon.

Succa 3:10

If one doesn’t know how to read hallel and a servant, lady or child is reading for them he must say the phrases himself. This is a curse that he does not know how to read himself. If it was a older person you answer halleluka on every sentence.