'Is it safe to visit London solo as a young female?' Why would it not be? With a population of about 8 million there are plenty of people your age and younger making their way through the capital every day, and very few come to harm. What is it you expect might be unsafe? (the answer to your question is yes, it's safe).
'Is it crazy to have my first flight ever be an international flight?' No, not at all. However, you may wish to fly with BA if possible to make the experience one in which the holiday will start from the time of boarding.
'Is 7-10 days too long/short a time for London and surrounding areas' Some may say that seven days is too long, for others 10 weeks will be a good start. If you visit for 10 days including arrival and departure days you will have just eight days on the ground, and a day trip out of London would be a very good addition to the trip.
'For this time duration trip, what would the necessary budget be? At first estimate I am guessing $3000-$4000?' The only part of your trip where the price will be shown and charged in $US is the airfare. For the rest, for your own benefit, please think in sterling. Use http://www.oanda.com to calculate the exchange rate. At a very rough estimate, goods that cost $US1 may cost £1 in London, bearing in mind that all taxes will be included, so the price at the shelf is the price you will pay at the till.
'Can you get by without having to rent a car? And would you fellow travelers consider the public transportation safe enough for me to rely on?' Yes and yes. Thousands of people who speak lilttle or no English manage on transport in London every day, so you will have one advantage before setting foot on a bus or into the Underground station. Paying for transport can be very easy as well, but that can wait for now.
'No dates picked yet, what is everyone's recommended time of year to visit London?' May is usually quite nice as is September, although there will be the first hints of autumn in the air at that time. June is also quite a nice time to visit, with very long days.
'Tips, hints, help in general for my first flight/international trip?' Do your homework as you've already started. London is a very large, crowded, busy city, but despite that there are areas where peace and quiet are attainable as well as plenty of things to see and do. In terms of your first flights, ensure you arrive at the airport in good time before departure, don't overpack and (if you haven't done this already) apply for a passport.
'Top things to see while in the London area?' Ultiimately that will be up to you. Guidebooks still have their uses, with the Rough Guides and Eyewitness Guides worthy of consideration. No two travellers will have the same wish list so you will have to seek out what appeals to you rather than follow someone else's plans.