You need to complete your LLC's tax return before you can complete your personal tax return, because you will need information from the LLC's tax return to complete your personal return. Much the same way you need to complete your federal return before you can complete your state return.
As far as when you file the returns with the IRS, I don't think filing your personal return before your business return is going to cause any major problems. However, I don't understand why this is coming up. You need to complete your business tax return before you can complete your personal return, so why not just file the business return after you have completed it? Why hold on to it? At the very least, why not just file them both on the same day? I can't think of a scenario where this issue would come up.
I don't know every single thing that would increase your chances of an audit, but I know that taking certain deductions will increase your chances. For example, taking the home office deduction.
Based on your mention of a K-1, I'm assuming a US based LLC with pass-through taxation.