To recap, you can't get rid of the router even fi you only use it as a modem, WiFi and more ports are an issue and from the sounds of it you don't have any fancy networking/routing/firewall requirements.
That being the case then a cheap 8 port unmanaged gigabit switch stuck into one of the SH LAN ports sorts out your port issue. Widely available everywhere for less than £20
That just leaves getting WiFi where you want it. The principle is you want something acting as a wireless access point (WAP) nearer the dead zones. But in order to broadcast WiFi whatever is in those dead zones requires the network from your SH to reach it. In order of my personal opinion of worst to best you have the following options:
- Wireless repeaters. I think this is what you have already tried. They are effectively a WAP and a wireless client in one. You put them half way between your router and the dead spot. They receive their connection to the router wirelessly and retransmit it. They are pants generally as you have found.
- Powerline. You use your mains cables effectively as extending your wired network. One plugs into a socket next to your router with an ethernet cable between the two. Then in theory all your mains cables are an extended network. Find an electrical socket near your deadspot and plug in an adapter that also acts as a WAP and in theory you've extended your WiFi range. However in practice they are dog slow, riddled with issues to watch out for, are dependent on the quality of your wiring, I've found can intermittently drop out and I don't like them.
- Mesh. The idea here is that you dot mesh units (also called nodes) around your house. The first one should be wired into your router with an ethernet cable. Subsequent nodes then talk back to this first node either directly or hop amongst themselves to further extend the reach. This can be via network cables for more expensive units or using their wireless. Personally I've never really understood why you'd spend more on wired backhaul connections for mesh nodes because I think you get a better solution from the next bullet point. If you really don't want to run network cables around your house then wireless mesh is my preferred method over the two above. This is what @peige has done in the example above.
- Wired infrastructure and WAPs. By far the best results because you're going to ensure you get gigabit going to all your WAPs. Here you run ethernet cables from your router/switch around the house to strategic points where you want WiFi. Where the cable ends you put a WAP to give you WiFi in that location and vicinity. Given you say you have plasterboard internal walls this isn't the biggest job. If you don't fancy it yourself most electricians will take it on. I also think it adds value to your house to have a nice neat wired network infrastructure in your home. This is conceptually what @233 has done in the example above.
In all cases (except repeaters where you'd leave WiFi on in the SH) you'd probably want to turn WiFi off on the router and let whichever solution be your sole provider of WiFi. The SH can remain as a router for its modem requirement to actually connect you to the internet and its routing/NAT/firewalling etc. that you're probably not interested in.
Having said that, it is still perfectly valid and sometimes desirable to take a step further as
@233 and
@peige have done and also remove routing duties from the SH leaving it to only act as a modem.
