IMac does not like being connected and shuts down

Question

A white 17-inch iMac belonging to relatives of mine has started to behave rather oddly. In particular it does not seem to like to be connected to anything. Once when set to run in Fire Wire–hard drive mode it had just shut down and restarted itself in normal mode. And now when I try to start it with an Ethernet cable connected it just turns black shortly after the start signal. If the cable is removed it starts as it should (most of the time anyway – sometimes I have had to pull the power plug for a while or do a PRAM reset).
In addition, once when I pressed the switch on the back to start it up there were repeated soughing sounds, as if it tried to start over and over again. When the power plug had been removed and reconnected the Imac started directly, without the switch being pressed.
Finally, once when I tried to wake it up from sleep mode it shut down too. What is wrong with it?

Answer 1

So, I tried the old ADSL modem and router again, and now the iMac seems to work fine and connects to the Internet without problems. (The reason this equipment was exchanged in the first place, some month or two ago, was that somebody felt that new technology should be used, something by means of which your telephone calls too are transmitted over the Internet. Well, obviously it did not work, at least not with this piece of junk that Telia (our previously national Swedish telephone company) sent.)
Now the only thing is that although the iMac seems to work fine obviously some damage has been done to it, since for example it shut down from target disk mode (without any Ethernet cable being attached). I do not really know how to address that.