To clarify your original question, by "small server", I'm guessing you don't mean server class hardware with additional features that let you simulate an attached CD-ROM or get remote console access out-of-band, correct? It's just a desktop PC being used as a server?
To answer, it's certainly possible to initiate the installation of a Linux distribution from an existing OS via ssh. I have done a similar thing a number of times, usually with Red Hat-derived distros, but the principles should apply here.
Copy the installation kernel and initrd to /boot of the old OS, edit the boot configuration (GRUB/LILO/whatever) so that the default is to boot the install kernel, along with arguments from the install media boot config (isolinux.cfg if it's a traditional CD or DVD), add in whatever arguments you need to allow the install init system to find the install media (in my case I usually did an NFS install). Once the installation routine is bootstrapped it is no different to an installation started from PXE/DVD/any other method.
Depending on boot arguments passed to the kernel/init/installer, you can either start an interactive install that you take control of somehow (I have used Red Hat's vnc installation feature [I have no idea if Debian offers a similar facility], usually connecting to a listening viewer running on my desktop; in your case, you coudl go and stand in front of the machine) or have the installation entirely scripted, such that once it boots from the installation kernel, you do not need to interact with it until it reboots into your newly installed OS. Be warned that with this method, if you are wiping out the old OS, should it fail after having repartitioned or mkfs'ed the /boot FS, you will be left with an un-bootable system, so I recommend testing any scripted installation thoroughly first.
In my case, this has saved me a great deal of time (much of the work for unattended installs already having been done), in your case, if it's just for one machine, you might find it less work to take another approach.
Regarding network booting, it's perfectly possible to have GRUB or LILO boot a gPXE image that then can do PXE on your NIC. Get a prebuilt .lkrn image from http://rom-o-matic.net/gpxe/gpxe-1.0.0/contrib/rom-o-matic/ . If you don't mind waiting, you could use the tiny gPXE image from http://boot.kernel.org/ to boot Debian Live (which I believe then lets you install to the hard disk) over the Internet.