I am married 3 years to my husband. My husband has a 10 year old son from a previous marriage that lasted 6 months. I am the 10 year old his Stepmom.
My husband and I have a two and a half year old daughter together. We want to get started on adopting my 10 year old stepson asap.
My husband has had temporary full custody of his son since he was 3 years old. His son has not seen his biological mother since he was three, nor has not contributed to our 10 year old financially or attempted to reach him. His biological mom is also remarried.

Basically my 10 year old and his biological mother would be strangers if they ever saw each other today.

Is this called abandonment? What are our chances of me adopting my stepson with out his biological mom's consent? Does a judge take into consideration my husband and I have a child together, and I have been the caretake for our 10 year old son full time since the age of 5.
Appreciate any and all feedback.

Sincerely,
Nesh

Your chances of adopting without her consent are effectively none. It CAN be done, but a termination of parental rights is almost never granted if there isn't crime involved. Just being non-supportive does not constitute grounds.

Have you tried to have child support enforced? Getting that rolling might encourage her to allow the adoption. Has she told you that she will not allow it?

Your current situation, other than being the step mother, factors little into the matter.

No the judge does not take any of those things into consideration basically, unless there is rights signed over by the other parent.

And of course that is part of the issue if you have not been forcing the other parent to pay child support, if they were having to pay they are often more willing to sign over rights to get out of paying.

But most states ( in the US) merely not seeing the child and not paying support is not enough to take away their rights.