Kenya - Rawanda - Tanzania?

I am looking to travel to all three of these countries from July 31 - August 19. IIn Kenya I am hoping to visit Nairobi and do a "safari" somewhere near Nairobi (in the south?) In Tanzania I'd like to climb Mt. K and visit Dar es Salam. In Rawanda I'd like to spend some time Kigali and see the gorillas (if possible) or the golden monkeys. I'd be flying into one of these countries from Cairo - I'd prefer to fly round trip into one location to save money so I'd have to do some kind of loop. I'd like to know:

1. If you think I have enough time to do all of this in just under three weeks. How long does Mt. K take?
2. Advice on traveling between the three countries - I'd prefer over land but I don't have much time. I was thinking of flying into Nairobi because it's the cheapest, so from there would it be best to head to Rawanda (how?), then Mt. K, then Dar, then back up to Nairobi through Mombassa (or some other city?) I have no idea how everything is connected - like is it possible to make my way from Kigali to Mt. K - and I'm not sure how long travel takes.
3. How much planning do I need to do in advance? Can I just show up in Nairobi/Kigali and plan activities/safaris from there, or should I reserve spots for things in advance? How far in advance?
4. I am a woman and I may be doing much of this alone. Will I be relatively safe - especially traveling by bus alone?

I'd appreciate any advice at all on these questions or anything else. Thank you!

the time frame is good and could make a great experience.it is the immigration time so thats guranteed to see thousands of wildbeest and much better if you catch them in the act of crossing the river, you may require at least 2 nights in the maasai mara alone and if you get to see other parks the better, so may be 6 to 7 days minimumly to get the best of kenya. You will also need 5 to 6 days climbing kilimanjaro, and dar is not too far from moshi after you climb, so thats a good drive as well although could also fly to Dar from arusha.

To see the Gorillas in Rawanda, you will require a permit, and require an advance booking and a schedule, either way can start with the Gorillas and fly to Nairobi, Dar or kilimanjaro to start the other way,and or vise.

Overlands mostly will do a particular schedule and not necessarily your intended plan,so may have to consider that, and i like your idea of flying to Nairobi and do the ground from there, then kilimanjaro, dar and the Gorillas, alternatively may have have to do the Gorillas first then by road fly in to Nairobi to continue from there. you can also access Dar from Mombasa by road and vise, as there are buses departing every day.
Depending on what budget you got, if camping then its not really that much advance booking although should not do a last minute thing,no,but if you intend to do a lodge safari- then you require pretty an advance booking due to availabilities and im sure you do not want to mess up your holiday with an overbooked lodge!i mean start now,if its nt late already!,
Travelling alone could be very expensive at times, and mostly if you are not so familiar with the region its not so advisable to just pop in. Madam, I suggest you take your time to do the homework now,so that you will feel even more safer when it comes to the real thing.

Good luck!