Is it possible to get all DNS records off a remote server?
Normally you can't, but if the DNS server allows zone tranfers to anyone (unlikely) you can do it.
Try Zonetransfer on unix shell:
$ dig axfr sld.tld. @nameserver
get a list of nameservers delegated for your zone:
$ dig soa sld.tld. +trace
Zone transfers are always available to slave nameservers. This will be at all the listed nameservers except the master nameserver. Higher security configuration hide the master nameserver and may not allow public access to it.
dig axfr sld.tld. @nameserver
Once upon a time, zone transfers were frequently available to everyone. Today's best practices discourage allowing them to everyone. This helps limit information leakage. The axfr command above will get the data if it is available.