steaks, the way Trump likes them...
You're right, "the way Trump likes them" is not a dish, but it is an explanatory phrase to clarify something else. The author is specifying that the steaks were prepared in the manner that Trump likes them. (I don't think this is an example of very good writing, by the way. It would have been much clearer if the author had written something like "steaks prepared the way Trump likes them".)
With some of the ink "mercifully well-ground" into the steak's chimichurri sauce
Ground is the correct word here, not grounded. This is the past participle of to grind: Today I grind the spices, yesterday I ground the spices. (Grounded would only be correct if the verb were to ground.) Again, this does not seem like the best writing to me - ink and chimichurri are both usually liquids, so I don't understand how you would "grind" ink into chimichurri (grinding is something you do to solids) but the meaning is clear enough: it is thoroughly and completely mixed in.
To wash it all down
"It" just refers to the food. To wash something down is an idiom that means "to have drink with your food"; to me it implies that there is a lot of food, so you would need to help it down with some liquid.