I'd definitely pull the starter and get it checked, sucks since it's new but if it's a reman it's very possible it wasn't right when you got it. Did it ever work or has it done this since you installed it? Double check and make sure all the teeth are present and not rounded on your flywheel. You might look into a vette starter like this one:
http://shop.advanceautoparts.com/webapp/wcs/stores/servlet/product_Starter-Worldwide_5849098-P_123_R|GRPSTRTAMS_539226739___
Just ask for a starter for a 96 vette, they're smaller and cheaper than the f-body starters.
As for push starting I've done it in 2nd, never tried it in 4th.