Yes. I've purchare and received lots of domain names from other domainer by my moniker's account, so I believe they do.
You will receive a email from Moniker to inform you that [xxx.com] domain name will push to your account. then Moniker will send you email everyday a "TRANSFER-IN STATUS REPORT" to let you trace the domain transfer - (the letter like this)
Yes if you mean push within moniker network. But if you push to other domain registrar,then you should pay the fee.