Remember that in this industry you do get what you pay for. There are a lot of things that can go wrong when buying a Property so you really do need someone who knows what they are doing.
You negotiate the settlement date on signing. Unless there is already one in the Contract. If you speak to the agent before the auction they should be able to give you an indication of when the Vendor wants to settle.
Make sure you make it clear to your solicitor that you really want to buy this Property so he doesn't jeopardise that. See if he can negotiate with the other solicitor first. Why can't you give them a particular date?