How to check if I can log in to server via ssh?

You can do that with a combination of the BatchMode option and "parsing" the output. (ssh always returns 255 if it fails to connect for whatever reason, so you can't use the return code to distinguish between types of failures.)

With BatchMode on, no password prompt or other interaction is attempted, so a connect that requires a password will fail. (I also put a ConnectTimeout in there which should be adjusted to fit your needs. And picked really bad filenames.)

#! /bin/bash

rm good no_auth other
while read ip host ; do
    status=$(ssh -o BatchMode=yes -o ConnectTimeout=5 $ip echo ok 2>&1)
    case $status in
        ok) echo $ip $host >> good ;;
        *"Permission denied"*) echo $ip $host $status >> no_auth ;;
        *) echo $ip $host $status >> other ;;
    esac
done < list.txt

You could detect other types of errors (like missing server public key) if you need more detailed classification. If you need the results in a single, sorted file, just cat the various output files together as you see fit.