Admin / Customer Emails not arriving

Is there anyway of changing the order sequence so that the store sends the order confirmation to the admin first, and then the customer, instead of the other way around?

We've had problems with email spam and are on several blacklists which are currently being sorted, but if a customer places an order from that type of email account, they don't receive their email, and hence neither do we, but the sale is logged in the orders section of the admin side of the site.

Or if a customer enters their email address wrong on sign-up, then place an order, again they don't receive it, and neither do we.

You've asked this twice before.
And the answer is: yes. You can customize the code any way you like. You'll have to reverse-engineer every instance where an email is sent and deconstruct the emails so that the admin email is built separately from the non-admin email, and then send the messages in reverse order. As it stands now the customer email is sent first and then admin stuff is appended to it and then the admin email copy is sent.
Further, when an email failure occurs, the store should carry on with sending the next email and finishing normal script execution. If yours isn't doing that, then either you've got it set to fail on email errors, instead of simply doing a friendly email failure message (Admin->Email Options), or your mailserver is the one who is actually rejecting the second email, presumably because it's confused by the other failing address as well.