One strategy I've found that works sometimes is, even if your initial offer is rejected, keep following the property you like after its status is changed to Pending.

The seller could then agree or reject your adjustment in price.

This process isn't supposed to be a giant mystery - if lenders / underwriters are rejecting you for a specific reason, you should be able to ask them what that reason is.

Probably submitted 10 offers since January, and 10 rejections.

because the listing agent made it seem like he was not in a hurry.My initial offer got rejected and I'm trying to structure a counter-offer with an incentive to the seller without bringing up the original offer price too much.

Bank accepts/rejects/counters the offerThey normally take about two weeks from time of offer to completion of appraisal/bpo.

We must accept the property after the inspection or we reject the property.  

By the next day I had an email from the lender cancelling there rejection of the previous offer and requesting I send out a new one with a close date within 30 days.

I put in a low ball, got rejected, it went to auction, and I picked it up during the post auction process before it goes back on MLS.

Also postcards get rejected and returned to sender so you can use that to save money on places that bounce, remove them off your list.3.