1- Many scholars believe that Imam Ali (a) was critical of these conquests and did not support them.
2- Imam Ali (a) did not participate in them because he did not find them to be legitimate. However, he sent some of his companions. Why? Probably so they can try to reduce the injustice committed by some Muslim fighters, or to play a good role in guiding those areas that came under conquest.
3- It’s not clear whether Imam Ali (a) encouraged Omar to fight the Persians. Some believe he did (since the Persians would later fight Muslims, so it was defensive in the sense that it was preemptive). Some doubt that.
4- We don’t have any authentic evidence that Imams Hassan (a) and Hussain (a) participated in those battles. Yes, other companions did participate.