Pretty much any ceiling mounted access point (AP) will give you massively better signal than you currently have. In my experience there was a significant improvement in mounting the AP on the ceiling in the upstairs room vs just laying it on the insulation in the loft space.
At first I though this was just down to the attic floor blocking the signal but apparently the access point needs a ground plane behind it to work optimally.
I personally have Ubiquiti Unifi kit and I can thoroughly recommend the newest version of the AP-AC-LR. This isn't the fastest Unifi access point on paper but it has more sensitive reception antennae so it gets you a signal on a phone or tablet more often than the AP-AC-Lite or AP-AC-Pro. Pro in theory is faster, but if you haven't got a signal, the extra speed is pointless. Make sure you get the newest version which is 48V PoE unlike the older LRs which really only work with Mikrotik or Ubiquiti switches that supply 24V Passive PoE. The individual APs do come with PoE injectors but a PoE switch makes installation so much easier.
If you're not that concerned with getting into the whole Unifi ecosystem then Mikrotik make a fantastic range of more conventionally managed APs. They do look VERY industrial though.
And yes, you will get WiFi from both the router and the AP although most people switch off the router transmitter as it just adds to the WiFi clutter in the area.
