WR1043ND + WRT54G for extending Wifi range. How to set them up to seamless signal switch

If you want to extend the wireless network wirelessly, you need to put them on the same channel. Otherwise, the extender won't be able to talk to both the base and its clients.

A WDS bridge won't work because a WDS bridge can't have its own clients. It only bridges wired clients. For WDS repeating, you don't have to use the same SSID, but it's recommended. It will make roaming work more seamlessly. For a client repeater, they must be on the same SSID or the extender won't become a client of the base.

The difference between a client repeater and a WDS repeater is that a WDS repeater does things right and a client repeater fakes it. You have to fake it if the base access point doesn't support WDS or for some reason you can't (or don't want to) configure it in the base access point. For example, if your base will disable 802.11n if you enable WDS, you may prefer not to enable it.

A WDS repeater acts much like a regular Ethernet bridge. Things should work just the same as if the clients were local. A client repeater has to impersonate its clients to the access point so the access point thinks they're its own wireless clients. This can break protocols other than IP and can break if you have more than one IP subnet on the network. However, for most simple configurations, it works fine.

Roaming from one access point to the other should work seamlessly with WDS. There may be a connectivity hiccup with a client repeater because the hardware address appears to change.

Yes, Apple's AirPort Express can join an arbitrary wireless network and extend it wirelessly with the same network name, and have it all act as a single bridged [W]LAN.

I seem to recall that some older "flat square" 802.11n AirPort Extremes and Time Capsules had a hidden way to do this as well, but that feature didn't make it into the current "tower" shaped 802.11ac models. But the current AirPort Express still does it.