Hello Sophia.
The issues raised by Rick Baggenstoss are a good place to start. You should also ask if they do the "entire enchilada!" (That's a technical mold-remediation term.) As in ... do they do the testing before-hand, then complete the remediation, then do the testing afterwards, to prove that they did the work correctly? It sounds good in principal, but it's actually a case of "the fox watching the hen house!"
In some states and localities, doing testing, remediation, and follow-up testing is not considered an issue. But do you really want a remediation contractor doing the work and then proving to you - via their own testing - that they did the work correctly. I doubt it.
Since your issue is so obvious, I would not recommend doing any air testing before the work is done. You've got mold - that's pretty obvious. If you wanted to have a single "tape lift" type of sample taken at one of the heaviest areas, you could. That would tell you what's there. That information may come in handy when the post-testing is done.
I think you just need to hire a remediation or water restoration company that knows how to remove it. Have them do their work. You might also ask them who they know that can do follow-up (clearance) testing. That should always be another company that has no financial or other interests in the remediation company. Reliable remediation companies should have no problem giving you the name of someone who can test and verify their work.
Do It Yourself
You could certainly do the demo before remediation, but you will have to notify everyone who comes into the house that there are high levels of mold airborne and on material surfaces. Even the hardiest of remodeling contractors can react to high levels of mold. Many do react just because they've been exposed to it in the past. Your choice here is related to your notifying people and your potential liability.
Again, you can follow Rick's advice "You should do a quick wipedown yourself / team, if you have one. Do your demo. Wear high quality respirators!" He's not talking about doing the detailed cleaning, but rather getting the bulk of the matter off of walls, ceilings, fixtures, furnishings, and perhaps vacuuming the carpet as well.
If you choose to do some level of cleaning yourself (prior to demo) then you can simply use mild dish-soap and water. Dawn or some other dish detergent. These soaps work exceptionally well at picking up large quantities of very small spores. The dish soap increases the surfactancy of the water (it makes the water wetter). This means smaller molecules of water work better at picking up the dirt, grime, dust, particles, and yes mold spores.
Good luck!
Travis West