Welcome to the Community,
Will it install (for the purpose of running just one slim VM) on the non-VT dual core 64bit CPU like an Intel E2200 with 2Gb RAM? Or do I *need* the VT in order to install?
You may be able to install ESXi, however you will not be able to run 64-bit guests without VT support. With only 2 GB RAM you will be able to install the OS and run it, but there will not be too much RAM left for the VM you want to run.
Can I install it on the headless system over the serial console?
Not sure what kind of console this is. As long as you can run the installation from the CD or attach the ISO as a virtual CD you should be able to install it. You will have to try it. If this does not work there may be other options. E.g. prepare an USB key with ESXi, attach it to the system and boot the Hypervisor from USB.
Whether or not it is possible to install and run ESXi on your system, not only depends on the CPU. There are other requirements for the hardware too. Like the disk controller, the network cards, ...
André