Is there a big difference between encoding the video and audio separately in the original video file and encoding the original video file as is?
If you're only downscaling the video, then you can save a small amount of time by separating the audio and converting it once, so for each uploaded video, there is 3 video encoding steps + 1 audio encoding step instead of 3 video encoding steps + 3 (redundant) audio encoding steps.
Here's an example of how to do this, taking $input and compressing it into three videos of 360p, 720p, and 1080p resolutions. It encodes the audio as 64kbps Opus. The video is encoded to AV1 with the default CRF:
ffmpeg -sn -dn -vn -i $input -c:a libopus -b:a 64k temp.opus
ffmpeg -sn -dn -an -i $input -i temp.opus -c:a copy -c:v libsvtav1 -vf "scale=-2:360:flags=lanczos+accurate_rnd" -c:a copy 360p.webm
ffmpeg -sn -dn -an -i $input -i temp.opus -c:a copy -c:v libsvtav1 -vf "scale=-2:720:flags=lanczos+accurate_rnd" -c:a copy 720p.webm
ffmpeg -sn -dn -an -i $input -i temp.opus -c:a copy -c:v libsvtav1 -vf "scale=-2:1080:flags=lanczos+accurate_rnd" -c:a copy 1080p.webm
rm temp.opus
There would be a way to do this all in a single command with no temporary files using filter_complex and the tee pseudo-muxer, but it would be more complex to read and isn't necessary to illustrate the point.
I want to change the resolution without compromising the original image quality.
That isn't possible. Whenever you transcode you damage quality. This is called generation loss and you should always minimize the number of generations of transcoding that a video goes through. When you downscale, you also remove information. You can optimize bitrate and resolution trade offs for best quality by using something called a "convex-hull approach".
Lastly, are there appropriate settings for the three resolutions (360, 720, 1080) that I am trying to convert to ffmpeg?
You should pick a CRF for each resolution that gives you acceptable quality at acceptable bitrate. Lower CRF means better quality but higher bitrate. You can also improve quality by using a slower preset. This will give better quality for any given bitrate (put another way, you can get the same quality at a lower bitrate) but at the expense of more encoding time. There are diminishing returns at the really slow presets though.