@Wisebird 

Hi,

Regarding your questions:

The phone is solely required for the initial setup process of the Range Extender and only if you opt to do the setup via TP-Link's app. This is completely unrelated to the strength of the Wi-Fi.
You can also do the setup via a web browser from any PC, phone, tablet (or even TV set, provided it has a modern web browser) that is connected to your local home network. In my opinion, using a web browser is the preferred method anyway, because the "Tether" app doesn't provide access to all the options that are available.

"Mesh" is not to be confused with "Powerline adapters", which use the electric wiring of a building to transfer data. Therefore, Powerline adapters would most likely not work for your scenario, because of separate electic circuits.
Whether the Mesh supported by the TP-Link devices would be usable in combination with your existing wireless router can't be said unless you let us know the brand and model name of your router.
However, TP-Link Range Extenders can still be used to extend the wireless network even if the "Mesh" feature is not supported by the router.

Of the two "Range Extenders" you've listed the one with the internal antennas appears to be the model RE330 and the other one with the external antennas should be the RE315. 
The antenna design seems to be the only difference between the two.

Besides the above, one important factor to consider for your case is, where is the wireless router located in relation to your neighbor's Firestick? (for example, 5 meters in horizontal → and 3 meters vetical ↓ )
Also, does your wireless router have external antennas that can be swiveled around or does it have internal antennas (this would be the case if the router doesn't have any tubes, poles or "ears" sticking out of its housing)?

  @woozle Many thanks for that. My router is a Huawei EG8145V5. Two attenae, one each side of the box. Three metres vertical and one metre horizontal from my neighbour, downstairs.

  @woozle  which device would you recommend, please?  I am grateful for your help.

  @Wisebird 

Ok. Apparently the EG8145V5 has two external antennas and two internal antennas, but I couldn't find any information about which one's are for 5 GHz and 2.4 GHz.

If your your neighbor's Fire TV stick is vertically offset by just one meter in relation to the router, then the Fire TV is likely in the dead spot of the router's wireless signal.
The thing is, common antennas can't spread their signal equally in every direction. As a compromise wireless routers are usually designed to spread most of the wireless signal sideways (provided they are placed as advertised, like standing with their feet on a flat surface and with the antennas upright), because that's what best fits most user's needs.

Before you decide to go for a range extender you might want to test if you can achieve satisfactory coverage of the Wi-Fi signal by changing the positioning of the two external antennas (by flipping them backwards) or the internal antennas (by tilting the whole router). Like in the sample pictures below.

Of course, doing this would likely reduce the coverage within your own apartment, but maybe it's still sufficient for you.

If you were to go for one of the two AC1200 range extenders, then perhaps the RE315 with the tiltable external antennas would be the better choice, as it would allow you to somewhat "steer" the signal towards your neighbor's Fire TV stick.

Also note that wireless range extenders (in general) are not a "cure-all". Not everyone who has bought them also ended up being satisfied with them. 

  @woozle Hi there.  I just wanted to say, thank you so much for all your advice and help.  It has been a real learning curve  for me.  I finally managed to the extender to my router.  I used WPS in the end. Then with the help of my neighbour, connected his TV and laptop. That bit I didn't know but he did.

Co-operation and sharing of knowledge.  Can't thank everyone, enough.  

Many thanks.