With the Edwardes pump, it is important that the condenser be above the pump so that the water can flow downhill all the way to the lowest point. The pump does actually pump both air and water, and this design is quite clever in that it uses the water to help give good seals on the piston and on the delivery valve. There is no foot valve, which is also quite a good feature, since these can be troublesome and are hard to get at. As noted above, the conical end of the pump plunger dispaces water up through the ports around the cylinder, so as the ports close on the upstroke there should be water lying on the piston, helping to give a good seal there. Then on the delivery side, the pipe to the hotwell should be uphill, and should deliver into the hot well over the top of the water level, so water will tend to lie on the delivery valve and help it to seal.
Full size plant usually does/did have a circulating pump. It is preferable if this can be driven separately from the main plant, but not vital. The worst effect is that it can take a little longer to establish vacuum if the condenser has got a bit hot.
I've seen it claimed before that the pump is not creating the vacuum. I'd like to point out that on starting, the condenser and exhaust spaces are liable to be full of air, and unless that is removed, a decent vacuum will not be obtained. Then once things are running, the pump has to remove the condensate, otherwise the whole system will become waterlogged. So while the reduction in volume by condensation obviously helps, there will never be a vacuum unless there is a pump there, or some equivalent like the above mentioned standpipes or a vacuum ejector.
Some beam engines used a jet condenser. This uses the vacuum to suck in a jet of water, sufficient to condense the steam, and a pump to remove the air and condensate. From experience watching a beam engine like this being started at MOTAT in Auckland, the vacuum tends to take a while to establish and then come in with a bit of a rush. While there is no vacuum, there is nothing to suck in the water, and the pump is trying to deal with a wet fog of steam and air. Then once a bit of water starts to get in, the steam condenses, and the pump suddenly has a much easier job.
John