Did you just graduate this year, or have you been working for a while? If the latter, the best advice I can give you, particularly at the district level, is to try to find someone you've worked with who knows the judge(s) you're targeting to put in a call for you. Unless the profs who are willing to go to bat for you are big names, I think a good word from a well-regarded local practitioner could get you further. As for how many, I think one should be enough to do the trick.
I graduated 2011 with a clerkship lined up for 2012, so not a lot of experience but didn't just graduate this year. For the judges that I'm targeting outside my local area, none of the profs or practitioners I'm thinking of would really know them; the closest I can get is someone that clerked in the same circuit.
It depends. Are you within spitting distance of getting an interview based on your qualifications, and not just geographical serendipity? My 9th Circuit judge (not nearly as remote as Alaska, but in a city that is out of the way for most applicants) would grant such an interview only if the applicant were already on the clerks' radar as a good prospect, or if a friend/colleague recommended the applicant. Federal judges are busy people and don't want to waste their time on interviewing tourists. You have to come up with something better than "I'm in town" to get your foot in the door someplace as competitive as the 9th Circuit.
I fully understand judges aren't interested in interviewing tourists. 9th Circuit and COA in general are always a little bit of a reach, but I believe my stats are at least within reasonable competitiveness, even if not at the top of their lists. I'm certainly never going to clerk on the Supreme Court or any feeder judges, but I've interviewed with a 9th Circuit judge in the past, and several district court judges, before getting the clerkship that I have now. My thought was basically along the lines of "this sounds like something that couldn't hurt and could only potentially help, even if it doesn't actually work out."
Also, does anyone else have any thoughts about doing a second district court clerkship? They're both for one year. I'm just worried that firms might question that choice.