Solaris

Not really, you can use straight cables for E150.
Btw, have the problem got resolved?

Its always good practice to use the correct cable type even when devices are smart enough to use either a straight through or crossover. It doesn't really matter if its an E150 or not . The things that will make a difference are the NIC itself, OBP, and the OS NIC driver. Also if its true that the machine in question can actually detect cable type (I have my doubts considering its age, I could be wrong though) and the cable type detection isn't working then the laptop test wouldn't be a good test in this case (I have seen strange hardware failures where certain components of a piece of hardware fail).

You should always use a crossover when you are connecting two NICs directly. The general rule with twisted pair cable types is like devices use a crossover, different devices use a straight through. Treat router interfaces like a NIC an a network node.