The p-trap is required by code. The only other traps that USED to be allowed and are grandfathered in are drum and u-traps, but you can’t use these in new construction. Your problem is fairly common one and is due to failure to provide for an ample tub box to provide trap positioning in the slab. As regards the vent, this needs to be placed in the drain line at the UPC specified distances. In a basement, that needs to be plumbed before the slab is placed, otherwise you will have problems requiring demolition.
NtP