I'm not specifcially familiar with Virginia or your benefits plans, but generally, yes, joint custody gives you equal footing as a legal guardian. It is a huge step up from CYA legal docs, and allows you to add the children to your insurance and pension, because if you are their guardian they are your dependents. It also allows you to do bunches of everyday things, like sign school permission slips.
It is harder to know what effect a failed application in VA would have, if any. It might depend on the reasons you were denied. For example, an investigating social worker found drugs in the house, etc.
Definitely, a lawyer can help here.