With visas, a tourist visa is for 30 days and will start on the date that you want it to start it on, so I would suggest that there is really no likelihood of it running out while you are there.
If the hotel you will be staying at provides visa support for free or at a cheap price then that is the best thing to go for, otherwise I would suggest using someone like Real Russia to get your visa for you:-
www.realrussia.co.uk/
There is absolutely nothing wrong with Aeroflot on international flights - they are a really good airline. I've had much better food and service with them than I ever had with KLM.
Easyjet have started flights to Moscow so that is a new option, whether you use them depends on what you think of them. If you aren't in the south-east then you don't have to go via London as there are plenty of flights from regional airports going with KLM, Air France, Lufthansa etc.
When you arrive, it is best to either pre-book a taxi or go to the central taxi booking point. Do not go with one of the touts - there are lots of them in the main Moscow airports and they can be very pushy.
Alternatively, there is a train service from both DME and SVO that goes into the centre - like the rail links from Heathrow and Gatwick.
Hotels can be very expensive indeed in Moscow. An alternative is to stay in an apartment. Here are two US/European owned agencies with English speaking staff and good feedback on Trip Advisor:-
www.moscowsuites.ru/
moscow-star.com/
When it comes to what to see that depends very much on your own likes/dislikes. Get hold of the Lonely Planet Moscow Guide and also the
Rough Guide to Moscow. Between them they do a very good job of covering just about everything that you might be interested in.
It might also be worthwhile taking a half or whole day tour. This company have a generally very good reputation:-
www.toursinrussia.com/
If there's anything else you need help with just give me a shout - I used to live there for three years.