Inferring from several things said in the rules, no - awaken does not confer a soul, since spells do only what they say they do. But it seems fairly clear to me at least that the rules assume living creatures already have a soul by default, which is why creatures like the ones you mention - zombies, golems and simulacra - must specifically be called out as not possessing a soul.

I base this on the fact that many important spells that interact with souls, like raise dead and clone, can target “a creature” - they are unrestricted by creature type or Intelligence score, and do not use a qualifier like “a creature with a soul”. (Some, like magic jar, do restrict some aspects of the spell to targeting humanoids or by creature size, but these match the constraints of many shape changing spells and can be assumed to be for other narrative and game balance reasons.)

This implies most creatures do have a soul, except those explicitly determined not to. (It would be reasonable to assume that creatures created by magic which does not say it grants a soul also do not have souls; e.g. animate objects or create homunculus, but they are very few such spells.)

To answer one of your sub questions: as defined in the introduction to the Monster Manual, regular plants are not creatures, but monsters with the plant type are:

Plants in this context are vegetable creatures, not ordinary flora.

Awaken does not confer a soul, since spells do only what they say they do. Inferring from several things said in the rules, though, the game assumes living creatures already have a soul by default, which is why creatures like the ones you mention - zombies, golems and simulacra - must specifically be called out as not possessing a soul.

I base this on the fact that many important spells that interact with souls, like raise dead and clone, can target “a creature” - they are unrestricted by creature type or Intelligence score, and do not use a qualifier like “a creature with a soul”. (Some, like magic jar, do restrict some aspects of the spell to targeting humanoids or by creature size, but these match the constraints of many shape changing spells and can be assumed to be for other narrative and game balance reasons.) Some still require a “willing soul”, and so can be vetoed by a DM based on the cosmology of their campaign or the desires of a specific target, but they are written to be inclusive.

This implies most creatures do have a soul, except those explicitly determined not to, and perhaps those created entirely by magic, such as with animate objects or create homunculus.

Writing about what happens to creatures after death focuses on humanoids, but this does not mean dogs and mind flayers and owl bears do not have souls, only that humanoids don’t think much about what happens to them when they die.

To answer one of your sub questions: as defined in the introduction to the Monster Manual, regular plants are not creatures, but monsters with the plant type are:

Plants in this context are vegetable creatures, not ordinary flora.