Yes, you can claim your child. You don't need any proof.
When two unmarried parents live together, with their child, either one may claim the child. They may decide between themselves which parent will claim the child. Only if they can't agree does the IRS have rules to set the priorities. It doesn't matter who provided the most support, or if the grandparents provided more support than the parents*.
If the grandparents want to claim the baby, instead of the parents, that is also allowed. But there is an additional rule. They can only do so if they have a more income than the parent with the highest income.
You may not claim your girlfriend as a dependent, unless you provide more support than her parents*. That is unlikely, if you live in their home.
*There are two types of dependents,
"Qualifying Children"(QC) and standard ("Qualifying
Relative" in IRS parlance even though they don't have to actually be
related). There is no income limit for a QC, and there is essentially no support test, but there is an age limit, a
relationship test and residence test. Only a QC qualifies a taxpayer for the
Earned Income Credit and the Child Tax Credit. They are interrelated but the
rules are different for each.
