Move compressed log files outside /var/log [logrotate with log2ram]

1. Yes, using the mv command without any options will overwrite any files with the same name. One option to prevent this is to use the --backup=numbered option, which creates a new file with the format filename~*[number]*~ instead of overwriting the existing file. Another option is to make use of other file naming formats like dateext, which renames rotated logs to filename-YYYYMMDD to avoid overwriting files with the same name.

2. olddir is the location where rotated logs should be stored. The logrotate man page warns against using wildcards, because if this option is absent, then rotated logs are stored in the same directory. If a wildcard is used to signal logrotate to rotate all files (*), then it would affect rotated logs too.

3. To define a task to be done at boot and shutdown, I suggest using either cronjobs or custom-written init services. Here is an example that should help fix your issue with logrotate:

/var/log/**/*.log {
    daily
    compress
    dateext
    missingok
    noifempty
    sharedscripts
    postrotate
        cd /var/log && find . -type f -name '*.gz' -exec rsync -aR {} /var/old.log/ \;
    endscript

A - global - preremove script can do it:

##  Archive rotated, compressed (`.gz`) log files to `/var/old.log/` before
##  `logrotate` removes them;
##
##    - Paths relative to `/var/log/` will be preserved in `/var/old.log/`;
##    - Paths outside of `/var/log/` produce an error;
##
##  Other files will be ignored;
##
preremove
    set -e
    case "${1}" in
        /var/log/*.gz)
            path_anchored=`echo "${1}" | sed -e 's!^\(/var/log/\)!\1./!'`
            rsync --archive --relative "${path_anchored}" /var/old.log/
            ;;
        *.gz)
            echo "Cannot archive \"${1}\"; path must start with \"/var/log/\"" >&2
            exit 1
            ;;
    esac
    exit 0
endscript

##  Make sure rotated log files have unique names, otherwise you'll be forever
##  archiving `file.log.7.gz` or whatever maximum count the `rotate` directive
##  has been set to.
##
dateext
dateformat .%Y-%m-%dU%s

You can add this as an entry in /etc/logrotate.d/- named such that it sorts before all other entries when /etc/logrotate.conf does include it.

This way, no other logrotate configs need to be altered, provided that they do not override preremove or set nodateext.

Any high rotate <count> will prevent rotated logs from being archived any time soon though, so those still might need tweaking.