To be clear, I am a moderator on both EL&U and ELL.
Mod messages are received in your global SE inbox (that's what appears when you click the Stack Exchange logo in the upper left corner of every page). If you click on the mod message there, you will be able to reply to it.
In general, we will respond to constructive queries; rants, taunts, and nonsense (all of which we have received in the past) will be ignored. In addition, you can ask that another moderator review the suspension.
If that channel of communication does not lead to a result satisfactory to you, you can contact the Stack Exchange team directly or post on meta once the suspension is complete.
The above is a somewhat general answer. I'm looking into the details of your second suspension right now, since it's not exactly clear what happened there, and I'll let you know what comes of it. It is perfectly fine to ask about your suspension, so we'll see if we can figure the details of that one out and get back to you.
I have sent you a mod message on ELL providing additional detail regarding the (second) suspension.