Your financing contingency, which includes appraisal, of course, needs to stay in place and would be expected.
Then there is the home inspection. It is not smart to waive the home inspection. For a purchase this large, you want to know what you are buying and any current or anticipated problems.
Our current home is under contract, with settlement scheduled in mid june. Would that be not sufficient enough to warrant removal of that said contingency, or is it removed only after the closing of the sale...?
In a multi-offer situ, they would autmatically throw that one out.
But if you are talking about "normal" contingencies; like Inspection, Loan and Appraisal; I would never recommend that my clients make an offer without those! Never!
If you got the acceptance, and then found out that the house was infected, or off it's foundation, what would you do then? You couldn't back out; you're stuck!
Do you think the other bidders are going without contingencies: If they are, I'd wish them good luck.