Would it be possible to mount a freewheel from an existing wheel on thie following type of wheel made for fixies? Or do I need a different axis for the rear wheel such as the one below?
The axle (the object on your picture) does not define which types of cogs can be used. It is the hub that rotates around that axle that determines it.
The flip-flop hub of the wheel you are linking to is designed to take a single speed freewheel (used also on BMX and some single speed city bicycles) on one side and a fixed cog on another side. It is not designed to work with multi-speed freewheels.
A simply longer axle is unlikely to help, as there are many other factors that define the compatibility. Among them are:
- The axle itself. Its diameter and bearing types/dimensions must match the hub body (i.e., if the hub has cartridge bearings, loose ball bearings and a cup-and-cone axle will not fit it.)
- Hub width. The fixie flip flop hub is 120 mm wide. Some of multispeed freewheels require 126 or 130 mm widths.
- Wheel dish. The multispeed wheel is more asymmetric because it has cogs only on one side, while a flip-flop hub has similar free spaces for cogs on both sides. A wrong wheel dish means the rim and the tire won't sit in the central plane of the bike but more on one side of it; as if that was not bad enough, it may also cause the tire to rub against the chainstay.
- Threading pitch of the freewheel must match the hub threads' pitch. That is something to keep in mind even when changing single-speed freewheels.
More about freewheels: https://www.sheldonbrown.com/freewheels.html