1) Does it have a separate Ethernet Card, or is it integrated?
(Part of the motherboard)
The Dell Optiplex GX620 comes with Integrated networking.
The chipset is a Broadcom 572.
It's part of the motherboard, and you plug the Ethernet Cable into the back of the computer.
However, you could have a separate Ethernet Card, plugged into the white PCI expansion slot on the motherboard.http://www.ebay.com/itm/Dell-Optiplex-GX620-SFF-Motherboard-F8101-KH290-PY423-20-Available-/370743464509?pt=Motherboards&hash=item565208523d
A) If you are using a separate Ethernet Card, replace it.
B) If you are using the Intergrated networking of the motherboard, (Plug Ethernet cable into back of computer), use an Ethernet Card instead.
White PCI slot.
You have a short circuit when you plug the Ethernet Cable into the motherboard. Don't use the motherboard networking, use an Ethernet Card.
2) Plugging an Ethernet Cable into a computer that is on, = No.
It is not a Plug N' Play item.
(Not stating this is what you are doing.
Just posting in case you are. Burns out the Ethernet chipset on the motherboard)
For additional questions please post in a Comment.