So first, your HIV is conclusive at 4 months. You do not have HIV.
Pregnancy totally messes with syphilis testing, so I'm hoping that they have confirmed all these tests with other tests. If you had a positive RPR, for example, they should do a confirmatory test after that to make sure these are accurate.
I can't tell you if you have neurosyphilis or not - only testing can determine that. It's not likely with secondary syphilis, as neurosyphilis happens when people have untreated syphilis for many, many years. Do you have an idea of when you were infected?
Please talk to your doctor about your post-partum anxiety. That is just as serious as post-partum depression. I would be surprised if syphilis is causing your anxiety, and not having just given birth, but your doctor can sort that out. Just talk to him/her about it. Whatever the cause, treatment in the form of counseling and/or meds can be really helpful. Remember that your mental health is just as important as your physical health.
You have so much going on, and I wish I had more definitive answers for you. Let's see what today's test says, and see if you can find out of the tests you've been given have been confirmed. This could just be a treatment failure, which is rare but does happen.
Hang in there and be kind to yourself.