As has been said, it seems there are no hard and fast rules. I've auditioned for others way more than I have auditioned others for me. But here are my thoughts:
As an auditioner:
Never audition 2 people at the same time. It's not fair to either participant to have to play off another person who may not be up to speed.
For sure, try to do all of your auditions for each role in one session. It can be brutal for a band to have to suffer through weeks of auditioning the same role and it's easier to compare your options when everything is fresh. For these that typically rehearse mid week, this can be a challenge because of time constraints, but keep the time frame as compact as possible.
Do allow buffer time between people. Allow at least 15 minutes to load in and 15 minutes to load out. Then another 15 minutes for the driveway to clear. Don't forget to include time to chat. Doing this during tear down is common, but there are distractions, so try to have a few minutes set aside to sit and chat - nothing else. This can be the most important part of the audition. Spacing multiple auditions at 90 minutes is about right.
Keep things moving. It is possible to have a relaxed, yet professional environment.
I like to pick out songs that demonstrate specific skills we're looking for, so I will specify which songs they need to have prepared. 4-6 is usually plenty. The audition should probably be different for each instrument. Songs that are good tests for the drummer are not necessarily good tests for the LS. I really don't care which songs they prefer. If there is enough time for a little jam after the formal audition, then you can explore any common ground.
Hopefully you have more than one candidate that could work. Be sure to deal with post audition communication professionally - don't burn your bridges.
Keep track of your secondary choices. It's not unheard of for a new member to abruptly quit (or be asked to leave) after a few weeks for any number of reasons. If nothing else you may be able to reach out to the second or third place person to sub for you - assuming the "sorry you didn't make it" news was delivered properly.
Make sure your band knows the material you're asking to be auditioned.
Don't just start grabbing stuff, but offer to help with the auditionee's load.
And maybe most important, be patient. It is tempting to settle for the person who is close enough wen nobody shined. Settling for the wrong person just drags out the process even further because you eventually have to start all over again. If you have a couple of "didn't knock me over but has potential" candidates, bring them back for a second audition with hopes that they were just having first date jitters for the first audition.
For both sides of the process:
It's about more then musical ability. Look for things about the other party that you like and things that you dislike. You are auditioning each other.
Be prompt and be ready. I waited over an hour at one audition because the LG was running late. This cut my session short because there as another waiting. (I still got the gig, but almost declined because of the late guy's lack of respect.)
You don't need to do the full audition. If things don't feel right and you know the fit is not good, it's just a waste of time to keep going.
Be appropriately groomed. I don't audition dressed to gig, but I am showered and wearing clean clothes. I expect the same of the other party.
Be as clear as you can regarding the administrative stuff before the audition is set. Discussions regarding rehearsal expectations, gig frequencies, travel?, pay structures, genre directions, strict cover copy?, etc can settle a lot of things before you even meet.
As an auditionee:
I like to see a band's full playlist, not just the audition songs. It gives me a better idea of what their game plan is. (I'll typically work up a good chunk of the full list - just in case there is an impromptu call out.)
Please have either a back line or at least an easy load in/out. (I guess this really doesn't apply to drummers and LS!) It really sucks having to dodge kids, dogs, expensive furniture or stairs when loading gear into someone's house.
Be clear in your communication. Don't make me guess what you want/mean.
If money is important, ask about typical gig pay.
Your biggest problem is that you are lacking a couple of key members and it is difficult for either party to audition without the other filled. I think in your situation, I'd be inclined to get the drummer in place, get them up to speed on your LS audition songs and then start the LS process.
Good Luck