I think you would be best served by a case that can hold at least 3 similarly sized drives for a useful RAID set up. This allows one drive failure and you retain 2/3rds of your total storage capacity.
I have been running a 2700X for a few years pn my media server. No GPU or integrated graphics. A processor is compeletely fine to transcode 1080p without a GPU as long as it is only 1 stream.
A bit on the software side:
A backup server is certainly not trivial to setup. If you don't want this thing to be a side hobby to sink 100s of hours into, just buy a synology NAS. It will be much smaller, more reliable, and easier to set everything up.
Do you know about containerization and docker? Tailscale + wireguard or reverse proxy + crowdsec + firewall setup? ZFS vs RAID?
With the stability of ZFS and its software raid and software raid support of BTRFS, there is not many reasons to use hardware RAID for a home server anymore. It has benefits in production, but you aren't planning on having 32 drives or anything.
Might be good to just install TrueNAS and make sure your security is set up correctly then use NextCloud and a VPN to backup from your phone. Simplest and most secure solution.
You have 3.5" drives in your build, but the case only supports 2.5" ones, so that's not going to fly.
The cooler should be fine for the cpu, and with the right drives, I'd say that setup would be fine.
Smaller cases will always have worse thermals, but it doesn't sound like you'll be driving it particularly hard. If you do end up reencoding, it could maybe-potentially put a crimp in your performance.