It seems Trump can't find anyone to take on his case. Trump Impeachment: President's Lawyers Back Away as Trial Looms - Bloomberg He would first need to pay Rudy for his past failures. It's not clear the government will pay for his defense so getting stiffed will be high up on the list for any firm. Can he use a public defender? Maybe just beg for mercy from the court?
Karl Rove: A Giuliani defense for Trump would 'raise the likelihood' of Republicans convicting him (msn.com) When clock runs out for Trump, Giuliani wants to play overtime | Newsday Giuliani uses unfounded 'Antifa' argument to defend Trump (msn.com) I agree with Rove. Bring on the mayor! Get a summons to Carone immediately.
Its an interesting question. The first question will be will Roberts preside? If he won't, how does the Senate replace him? What authority will their 'ruling' have on Trump?
You're right, it will be Constitutional trail, not an unconstitutional trial, because the Senate has sole authority over impeachment trials ... and yes, the Senate has impeached a private citizen before (1875, Secretary of War).
He should get Rudy and the crazy "Hugo Chaves conspiracy" lady. That should make the trial particularly fun to watch.