Yom Ha'atzma'ut - Israel Independence Day occurs on 5 or, as is this year, 6 Iyyar. This date is during the mourning period of sefirah according to both common minhagim (whether you begin on 2nd day of Pesach or Rosh Hodesh Iyyar.) if so, how is playing and listening to live music allowed as I see many religious Jews doing?
For those who do so, saying Hallel, and therefore rejoicing, on Yom Ha'atzma'ut is either a mitzva (e.g. according to R' Meshulem Roth) or at least a minhag.
Mourning during the Omer is a minhag. The aspect of it that prohibits music generally, as opposed to weddings specifically, is a relatively recent minhag.
Therefore, rejoicing on Yom Ha'atzma'ut overrides mourning during the Omer either because a mitzva overrides a minhag or because a minhag can override another minhag, and rejoicing over the more recent event overrides mourning the more distant event.
I recommend that you read the whole essay for more detail on this and related topics.