If the Dell computer is truly a new one and not just new to you, then it probably won't boot with the old hard drive. If it's a used model, it may boot, but you won't know until you try. If you want to try it, I recommend that you back up your important data from that old computer first. Windows 7 has been out of support for quite a while, it's time to switch to Windows 10.

On the other hand, a bad ethernet port doesn't mean the end for the old computer. You should be able to add an ethernet adapter, either by an internal slot if it's a desktop computer, or by a USB port. But you really should move away from Windows 7.