I was there in early August for 3 days. The cheapest way is to go by Tiger Air to udon Thani, take a shutter to the central Bus Station in town and catch the official air-con bus to Vientaine. The bus will stop and wait for you at both the Thai and Loas Immigration. However you have to have either a pre-acquired visa or from one of the ASEAN countries! The trip, which took us about one and a half hours will bring you to the Bus Station behind the Vientaine Morning Market. I would suggest you find a hotel nearby. I can tell you what I found during the trip if you are keen.
I think it is safe for you, but the normal advise of avoiding secluded areas during nightfall applies.
I think 3 days is too short as the journey to and fro Udon Thani takes up half a day. One week will be just nice for you to go up country to see more!
no place is all safe for a girl alone, although some place is safer than the other. when we went, there is a 20+ american guy who was missing with the poster up saying where he was last seen (a few months before we saw the poster). also have known of someone who travelled alone to cambodia and was found deceased for a few days in her hotel room. no investigation proceeds. likewise a guy went to take waterfall in yunnan alone... and have mishap...
i would say, at least have 1 companion. at least two more companions would be good. one can still go alone and most are safe though.
also depends on how you travel. it is best to always be in places where there is a lot of people, which is easier in busy towns and cities. however, if you take a tut tut alone in roads with no one around, anything can happened. some risk taking is needed, even for guys.
laos, i would say should be more safe than most 3rd world countries and crime-ridden 1st world cities....
seek more advice from wanderingcat at sgtravellors, think she can tell you more from a girl's perspective. you are also more likely to get female companions there, although they are less likely to be photographers.
cost wise is very cheap. double costs US$5-15 for budgets. food and transport also cheap too. $500 (exclude flight) is more than enough, and if you dun buy lots of things at luang prabang, you are likely to bring some money back.