There's no such thing as a renewal;, you need to male a new application.
Parents should provide proof of their personal funds, not yours.
Invitation letter is not a requirement for TRV.
You are in a tough and understandable situation but the one who has full custody has all the cards unfortunately and decides what's best for the children. You need to start with a custody issue.
1. Send a Use of a Representative Form (IMM 5476) via webform.
2. The lawyer's access will be revoked.
3. You can link it in the express entry account.
It might be okay. But do attach your explanation, show your efforts to get a corrected birth certificate and provide some estimate when you can get it.
Agreed AOR arrives next day tops.
You can apply for BOWP after AOR, but IRCC will start processing BOWP only after your PR application completeness check is done (that;'s about when you get biometrics request, 1-3 months).