If you have a chance to obtain a client for yourself or your firm, do so immediately. Don't wait for a second chance, someone else will grab the client.   This is the key to your firm's survival AND to your success in the firm. I'm not the best person to tell you how to get clients for a firm because I have been a sole practitioner most of my career, but here's how I would approach it: 

(1) When people ask you what you do, do NOT answer: "I'm a lawyer." 

DO answer: I'm a lawyer, I specialize in representing purple people eaters (or if you think you are too much of a newby to claim a specialty), I work for the purple people eater section of Smith, Jones, and Stickem.  (If you mention your specialty, you are more likely to get the attention of purple people eaters). 

If people ask you for a further explanation of what you do, give them an explantion which will serve as a sixty second pitch for your firm ("I am really enthused about my job because I am working with such great people and such talented lawyers. We were the first firm in this area to establish a purple people eater section and we have eight lawyers who specialize in the problems of purple people eaters. Blah, blah. "

(2) Get your own business cards if you don't have them already and ask your boss for some of his business cards.   Explain to your boss that you occasionally meet people (through your hobby, through your husband's work, whatever) who might want to be clients and you want to be ready to discuss the firm's services with them in hope of reeling them in.   Ask him for suggestions on how to do this. 

Sample conversation:   PPE:   I need info on PPE tax problems. (Follows with long complicated question you have no idea how to answer). 

YOU:  I'm really not experienced enough to respond to your questions.  Here, let me give you my boss's business card, he specializes in this (brief description of how great your boss is).  I'm going to write my name on the back.  I'll let him know to expect your call and I'm sure he'll be happy to give you a 5-10 minute phone conference at no charge. I'll let him know to expect your call - when you talk to him, please mention that I'm the person who suggested you call him and that way he'll know you're the person he's expecting and he'll have some idea what it's about." 

Then let your boss know about the conversation and ask if you could have done anything differently/better.

PS.  If you do continue to hang back and wait for second chances with prospective clients, I would be very interested to know what state and city you are in and please supply a list of clients. Because if you will give me two weeks to figure out how to approach these people, they're gonna be MY clients.  The race is to the swift here.,,

I see I haven't really answered your specific questions.

(1) Billing - if the case is good, almost any law firm will be interested in working out the type of billing arrangement that the client requests, and if they aren't they can refer the case to another firm and get a referral fee.  So you should probably ask your boss in general what your firm's billing policies are, but if a prospective client with a good-sounding case needs a lawyer, the billing arrangement is almost never going to be an issue.  The right answer is almost always, "I think our firm makes that type of billing arrangement if the case is solid.  You would need to talk to one of our partners to see exactly what they might be able to work out with you." 

(2) Conflicts -  This has never been an issue for me because, as a sole practitioner, I nearly always know immediately if I have a conflict.  My impression is that most large firms have conflicts policies and people who are officially in charge of screening for conflicts and you need to find out exactly what your firm's policies are and how these matters are handled. 

I was pondering your problem while I drove my kids to school and I came up with a few final thoughts - so you are getting installment 3 of a long, rambling, and possibly not very helpful answer (maybe other lawyers will benefit even if you don't). 

Young lawyers don't have good instincts about screening clients.  If the lost heir to the throne of Russia shows up wanting a law firm to file suit to get him restored to his throne (on a contingency basis, of course, with a substantial advance to the client to cover living expenses), there's always gonna be some young associate who wants to take the case.   The associates who bring in these kinds of clients are guilty of two serious crimes: showing poor judgment, wasting a partner's time.   At worst, the associate convinces the firm to take the case and the firm loses money, this is NOT good.

On the other hand, young lawyers who don't bring in business (especially young female lawyers) are at very high risk of being relegated to a second class tier as "permanent helpers." 

So young lawyers have to be very careful about bringing in business and they also have to be very careful about not bringing in business. 

My gut reaction is, if an associate - male or female - doesn't get coaching on this issue, it's important to ask for coaching, and this goes double for a female associate because this is the point at which subtle but eventually teminal sex discrimination can begin to set in and strangle the female associate's career. 

Thank you so much for some really good and useful advice.  You know I definately get the impression that my firm expects young associates to just work (and work and work) and to really not be at all concerned about brining in business.  However, I think this concept is based on the notion that all young associates are 25-27 years old and don't really know anybody except their family and college friends (neither of which does the firm want as clients).    Not to mention the fact that these young people don't really know how to work and learning that takes a lot of energy.  In my case, I am older and have numerous connections to business people in my city through a variety of organizations and networks.  I often find myself in social situations that could turn into rainmaking opportunities.

 I am going to have to seek out some guidance on this, and soon, before you come grab these clients :).

Thanks!