Can I? If not is it because in the low energy mode the network card or the hard drives shutdown? If they do is there any way to prevent this (ie. with a custom low energy mode)?
In full sleep mode, there is not a way to prevent this, exactly for the reason you listed.
Your best bet is to make a power profile to turn off the monitors after a certain amount of time. Just about every other component in the machine needs to stay on to keep the download going except for optical drives and such. The PSU has to keep the drives spinning, while they communicate with the motherboard (which may or may not have your Ethernet integrated, but still must stay on nonetheless), which will communicate with your memory which holds a buffer containing the program doing the downloading.
If power consumption is a big issue, you could purchase ( or build ) a small NAS device and load FreeNAS on it. You can use tools such as wget to download files to this storage device while consuming very little power.