When is the best time of year to travel to Tokyo?
Which airline is recommended? How much should I expect to pay flying out of LAX?
Any time is a good time for Tokyo but during Sakura is awesome. If you can, fly into Haneda which is way closer to Tokyo proper than Narita. I've seen flights as low as a $500 when there are sales.
I would fly ANA from San Jose to Tokyo Narita. San Jose's airport is nowhere near the mess of LAX, which is great for airport security, check-in and getting to your gate. ANA also uses a 787 for the flight, which is one really nice airplane to fly in. Of course, you live in LA, I think ... so that might change things for you.
Time of year depends. I went in October, and it was so nice. But you might want to go during sumo season. It really depends on what sort of activities are high on your list. The October weather was perfect. But no sumo for me!
I think weather-wise, Tokyo (and Japan in general) is at it best in Spring (sakura cherry blossoms and other flowers, people start going out after the cold period) and Autumn (great food, beautiful fall foliage in Japanese gardens).
Winter can be nice though it might get cold. Then again, you can enjoy onsen hot springs in the snow, various snow festivals, great skiing in Nagano or Hokkaido. Summer is very hot, humid and oppressive, and then you have the annoying rainy season. OTOH that's also the season of interesting Summer festivals and awesome beer gardens.
All that being said, Tokyo is a big city and you'll always find something interesting in the city and its surroundings (I do even after 10 years of living here).
Like Justin mentioned, it really depends on what activities you want to do during your trip. I chose late March through late April for my trip because I wanted to catch the cherry blossoms in full bloom. Everyone seems to be outside during this time, picnicking under the tress while the blossoms fall off in a way that makes it seem like it's snowing. In Tokyo you can walk along the river that runs through Nakameguro and it will be completely covered with pink petals. It's gorgeous!
The weather is milder during this time as well, never getting too hot or too cold. The summer can be unbearable and the winter too cold. So unless either of those temperature options sound good to you, I'd recommend going in the Spring or Fall.
I will admit though, I really want to experience Hokkaido in the winter, so my next trip to Japan will have to include that.
As for airlines, if you are a FF on any airline, I would recommend going with a partner airline that flies to Japan so that you can continue to rack up miles. Now that Haneda is open, I would definitely fly into that airport if you can since it's so much closer to Tokyo, but I suspect Narita still has a high volume of traffic and therefore might have better fare deals. The train from Narita to Tokyo is great, comfortable, and fast, so it's really not bad if you have to fly into Narita and take the train.