More specifically, I am curious if/how the timeliness of the Freddie Mac's response and the amount of the counteroffer in relation to your offer influences your response to their counteroffer.  

Even a Hud counter offer is not really a counter at all, more of an invitation to rebid for the next day bid openingsTypically an auction starts with a low price and is bid upwards often with the bidders aware of the current price and have an option to up the price. 

@Aaron Mazzrillo I agree, I call that fishing for counter offers.

I'm sure there will be a counter offer so start low and see where it goes is my thoughts on it. 

Originally listed at $3.7M, I was able to negotiate the price down (on several back and forth counter offers) to $2.7M, successfully reducing the price by $1,000.000!

Just a developing pet peeve -- cannot stand it when an agent tells me, Well, the extra thousands they want you to pay in their counter-offer would only be $X more a month, and that's not much at all!

I was told that the process to obtain an answer or counteroffer would take 60 days.

This is where I think that it could get tricky.If they accept your offer with the clause included it should be ok, BUTif they counter your offer then they can claim via the acceptance offer that they removed the clause via the counter offer.

Two days later, I submit a counter-offer of $57K all-cash, without the stipulation that says I can switch to financing if I want.

I am waiting for the hard copy of the counter offer so I can re-counter and hopefully meet in the middle.Any advice?