English Grammar

1. as ABC company is singular, of there the second clause should be expressed in singular as well. i.e. just use 'she', otherwise the 2 clauses will be in conflict with each other.
2. for singular verb in front, singular verb should follow. i.e. use 'she'. If people say 'they start produciton today', then people can say 'their workers are on srike now. The 2 clauses should match each other even in conversation.
However, for your info, sometimes people may speak of wrong grammer, and people just don't care of it.
Just like we speak Chinese, you may aware sometimes people use wrong Chinese grammer, as far as we can understand, we just don't care if the Chinese grammer is correct or wrong.
But it is more important to use correct grammer in writing, especially for formal cases.

Though ABC Company starts production today, all the workers are launching a strike now.

The example is a bit strange. I would probably say: Although ABC company starts their production today, all the workers are currently on strike.

Once you have eventhough or although, you don't need but or however etc. Those are just pointless. Their or her is also unecessary because the sentence follows the previous statement, and most people should understand!