Sonarr and Radarr are essentially tailored webpages. You will probably not get both running on one Pi Zero W nicely, due to ram constraints (A Pi Zero 2W would be okay). However One of those apps on a Pi Zero W would be fine, but given enough swap its possible that both would run (albeit slowish, but who cares? its automated so you don’t have to! Doesn’t matter if the “grab” takes a couple more seconds than normally)
There would be a couple of things to note, firstly a slim starting distro, and remove stuff unnecessary. Secondly, you would go the nginx route rather than Apache2 since it is far more lightweight and less RAM overhead. Finally make sure you have a swap area on your memory (preferably not on SD, but you have little choice) The OS (assuming Debian or whatever) will be able to swap out other processes when memory gets low for the active process.
Finally, transfer would be fine using wifi, alongside the browsing data but this is likely to stutter on occasions when you are browsing whilst it is transferring an NZB etc to nzbget or whatever your dldr is. (NZB’s these days can be quite large, especially for a 60GB BR rip type grab.) For torrents you’d probably not notice.
The PI Zero 2 W, is essentially a RPi3b with less ram but with multicores, so it would be more competent (due to being about to switch out to swap faster/more efficiently). But there is no reason why Pi Zero W could not run one of those apps (maybe both, depending on the local db sizes). After all, they are only single threaded anyway, its just how the OS works with them.
(Most of the posts in this thread obviously have little to no knowledge of how Linux works, or the capabilities of the hardware.)