The NT-505 will work just fine with a WiFi extender providing the extender is getting a clean and constant signal from your router which based on your photos and comments is in another room. How far away from the router is the extender and in what direction. Also how many walls are between the router and extender? If the extender is not getting a clean and constant signal from the router is won't matter which brand of streamer you replace it with. I ran an NT-505 for over three years in two different rooms with an extender but in both cases the extender had a reasonable line of sight to the router. Ultimately your only clean solution may be to pay someone to run a line from the room where your router is to behind your audio gear with wall jacks on both ends.
Also if you want to use a DDC to convert and output from a streamer to a DAC then you need to buy a DAC with a standard I2s input like Denafrips, Holo and others and not RJ45 as that is likely proprietary to Sonnet.