Hi Sian and welcome to travelpunk.com !
That's a great project, that requires some planning, I agree ! As Tumblezweedz wrote, I spent a few months in Nepal, and although I didn't trek the Himalayas I was hanging out in Kathmandu with a bunch of workers of the local tourist industry. I saw lots of travellers leave for a trek frantic, and come back extatic !
I do have a few questions for you, though. You say you're trekking "Everest up until mid december"... I assume you mean the Everest base camp, otherwise you need years of prep and a huge team to support you (and $50 000) to climb the mountain all the way up. Am I correct ? And "until December" means you want to start when ? The duration of the journey depends on the starting point : from Jiri (3 weeks) or Lukla (2 weeks).
« I am a female travelling alone and wanted to make my own way up to everest base camp staying in tea houses along the way - is this a good idea?»
It's feasable, with a good prep setting the exact itinerary, etc. However I don't think it's a good idea for security matters -the maoists and the health factor, you'll be above 3000 meters even in the trek prior to the Everest Base Camp-. To have one or more Nepalis with you, or at the very least trek mates, will help dealing with those issues. The porters adn trekking guides are used to face the maioists, and know how to deal with them, usually they ask for a $100 "contribution", and give you a receipt. But the recent political changes in the country may have solved that problem, I don't know. One important thing to know, seducing clients is a sport for trekking guides...
«I wanted to join an organised tour to climb one of the peaks, but only from everest base camp onwards, i didn't want to have to follow a tour the whole way up, is this possible?»
Well, what I said just before. thousands of travellers trek in the Himalayas every year with no organisation. Most do use the services of porters and/or guides through trekking agencies, small groups some time go alone (never saw somebody totally alone though). For such a long trek, don't go alone. Maybe if you really don't want a nepalese with you, try to hook up with other travellers. In Thamel, the tourist district of Kathmandu, you'll meet them.
And what do you mean by «climb one of the peaks» ? There's only one Everest ?
Or are you interested in other peaks such as the Annapurna ?
«Northern India for the month before - is it worth buying a warm coat and sleeping bag before i go? I have heard mixed reviews about what can be bought in Katmandu and its quality. Also what should I take for my bottom half, i am a bit worried i am going to be to cold?»
Two questions to answer you properly. Are you first going to Kathmandu then south to India and back up north ? Which part of Northern India will you be going to ? Under Nepal or the himalayan area of India ?
«Travel overland into Tibet - is this possible?» It is unfortunately impossible. All you can do is an organised tour from/to Kathmandu. The only way to enter is to fly to China. I was in Nepal in January, and there were organised tours and flights. But often, not the cold but the thick fog would cancel activities, and of course maiost strikes !
Don't hesistate for more infos !
, I have no answers to your questions...SORRY...but i had to say that I think your plans are absolutely wicked. I travelled alone in Europe last year and everyone I met thought i had guts to do it!!! Your trip sounds incredible and something I definately want to do someday...its inspiring to hear someone with your ambition. 
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