Is it permissible for a husband to tie his hair with a rubber band during ihram for Hajj, knowing that he will shave it? And is shaving better for Umrah, or is shortening better, given that he will shave during Hajj, especially since he will perform Hajj at-tamattu'?

If a husband performs Hajj mutamatti'an (combining 'Umrah and Hajj), it is preferable for him to shorten his hair after 'Umrah and reserve the complete shaving/trimming for Hajj. As for what is used to tie the head, if it is a thread or similar, there is no harm in it, because it is not considered a head covering. A muhrim (one in the state of ihram) is prohibited from covering his head with something that is customarily considered a covering, such such as a turban, clay, or thick henna. There is no harm in tying the head with a thread or similar.