1- - Who will I specifically be working with on a day to day basis?
It is often the case that a Principal will be responsible for winning projects and making the pitch while subordinate staff will be responsible for actually doing the work. If you want the Principal to be the one working on the project, make sure that is clear from the start so the job can be priced appropriately. Architects and Designers are so conditioned to be competitively priced that the Principal's time is not factored largely into each project.
2- Have you completed any recent projects on a budget similar to mine?
I was once asked if I ever had a project where there was no budget and the Client gave me total freedom to design. I assumed it was a joke. Every client has a budget and if you have a tight budget, you want someone who is used to working with tight budgets.
3- How much of my involvement will be necessary or allowed?
I have never had a client without an opinion. Clients like to be involved, and you are probably no different. Some Architects are great collaborators, and some liked to be left alone to work their magic. If you are hiring an Architect or Designer with a specific 'style' they may not take kindly to collaborating so make sure you know where you stand up front.
4- If the project goes over budget, what responsibility do you have?
This is the single biggest problem I have seen in the Design industry. Designers and Architects don't have control over the construction pricing but have designed something that is far from the reality of the Client's budget. Often it is late in the project when this is realized and then the Client has to pay for additional services to 're-design' to the budget. I might be in the minority here, but if your Client is relying on you to Design to a budget then you have to take responsibility for the pricing. As a Design professional you should be familiar enough with relative costs to present options that the Client can afford. If your Architect doesn't want any part of this, it may be a red flag when the change orders start rolling in.
5- Can I call your previous clients?
No matter what any of us say about our own work, the words of our Clients form our reputation. Speaking candidly with previous clients will give you a sense of what to expect on your project. If possible, try to contact clients that you may be aware of rather than the ones that are given to you.
Any large project brings its own stresses with it, but by asking the right questions up front, you can be prepared to deal with what lies ahead. Good Luck!