I have three older sons from a previous relationship.
She has 1 son from a previous relationship.
We have a child together who is 1 1/2.
We have been living together for 3 years with minor problems.
We were engaged and planning our lives together future.
I love my son and miss him dearly
I'm not abussive and live spending precious time with him.
He makes me happy.

Is she wrong to tell me to sign over my rights to him, having nothing else to say but "I hate your other three sons." And "you don't make enough money."
??

Laugh at her, there is no such thing as signing over your rights, it is a myth.

Tell her that you want joint custody and that you will see her in court

I don't believe that there have been only 'minor problems' during the three years you've been together.

How old are all these children (other than the 1 1/2 year old)?

Why do you reduce the reasons for her suddenly kicking you out and demanding you sign over your parental rights, down to "I hate your other three sons." And "you don't make enough money."

What are the not so 'minor' issues going on here?

Only a court can terminate parental rights. Courts are very reluctant to do so. So you can't sign over your rights even if you want to.

She can tell you to get out, but she can't just kick you out. Even if she owns the property or is the sole lease, this is your residence too and she would have to formally evict you.

She can tell you anything. But since you posted this in the Family Law forum, the legal answer is she can't enforce anything without going to court.