External antennas of same router on different floors

Does the router communicate with the same wifi-device through more than one antenna? (which obviously would not work anymore in case the antennas are all on different floors/stories.

Often, yes – features like beamforming (for MIMO) or STBC use multiple antennas at once. I think the calculations even assume a specific distance between the antennas in order to achieve this.

So don't do that. It might still work somewhat, but I expect that it'll really make the performance worse instead of improving it.

If you can run a cable, better run a regular Ethernet cable – connect it to a standalone Wi-Fi access point (or "wireless extender" as they get called nowadays) and configure both to have the same network name (SSID).

This will have many advantages over your plan:

• you don't need to worry about signal loss which you'd get with long antenna cables (see comments below);
• you can have as many APs as you need (not limited to just 2 antenna ports);
• each AP can use a different radio channel (and should, to avoid interference) allowing more devices to speak at once;
• you can reuse the same Ethernet uplink for APs and other wired devices;
• should the TL-WDR4300 or any AP die, replacing it will be easier (no need to worry about the specific kind of antenna connection).

Many APs can accept PoE (Power over Ethernet), although not all follow the same standard – but you can still put a PoE-powered AP anywhere you can drag an Ethernet cable, even if there's no mains access.

Side note 1: If you have a spare combo wireless router, it can be configured to act as an AP/extender (bridge mode) – this site has plenty of posts about that.

Side note 2: In places which neither Wi-Fi nor an Ethernet cable can reach, worst case you could even connect an AP through powerline (many HomePlug adapters even have built-in Wi-Fi).