Oh wow, so work can begin before the contract is signed? Seems like it could take forever for me to find out who won then.
Yep. If you want to do business in the federal space as a prime contractor, you have to work at risk. There's no other way to meet the deadlines in the contracts. What really sucks is starting work, working for six months without getting paid, and then hitting some roadblock that forces the winner to give up the work. This is rare but I've seen it happen with big companies that have inflexible contracts people. They never get paid for that and sometimes the subs get screwed, too. The government will then go to the #2 on the list and start from scratch.
I worry that if I show interest then they will think I'm eager. But I want to give the vibe that I have so much going on and things are going so well with my job and I have so many other job offers that I don't need them, and make me seem highly desired, and worth paying a premium for.
I've been doing this for a while, and I'd like to encourage you to reconsider your approach to playing hard to get. Showing interest in taking a job does not reduce your value. If anything, it shows your employer that you are invested and interested in the work, which raises your value. They don't want to replace you. If you pretend to not care about the job, couldn't they interpret that as a lack of interest? If you pretend to be too busy for them, couldn't they interpret that as a need to look for someone else who wanted the job?
I recommend you call them up today, let them know you're still interested in the contract, and ask them if they have any news on the award. Let them know you'll call again in a month to check in. Say THANK YOU for the opportunity to work with them on this deal.
Regarding rates: I like to agree on a rate up front, to help the company put together their bid responsibly and understand their cost. When it comes to starting work, you'll probably be asked to reduce that rate as part of the Best and Final Offer (BAFO) process. You can decide to do it or not. Some RFP's require monetary penalties for changing named staff. Ultimately, you need to decide what your price is and negotiate it as best you can.
Keep in mind that your leverage is limited to their rate to the customer. If they bid you at $80/h, then agree to $75 during the BAFO, you're unlikely to get more than $45-50, for example. It helps to know their margin guidelines. I use 40% as a standard margin when I lack other info. The financials are usually hidden but there are ways to ferret them out. One way is to ask them. Some companies are very honest about their margin requirements.