For the past few years, I've been making regular trips to Japan around the July/August and December/January timeframes. Unfortunately, for many reasons, I probably won't be able to continue that trend for this year. But, I might be able to help answer some of your questions.
Flights from the US always go to Narita or Haneda, so those are really the only airports you should look at. Airfare from the US to Japan sucks... period. It's especially bad during the July/August and December/January timeframes, because those are the times of the year when most people take vacations. I've paid close to $2000 for a round-trip ticket to Tokyo before, so $1400 for a July ticket actually isn't that bad. The only thing you can really do is scour the Internet for deals. In addition to checking sites like Kayak, Priceline, and Orbitz, check the websites of the airlines, as they may actually have better deals. Travel agencies might also have some specials, but that doesn't happen very often. One other thing I can recommend only applies if you live near an international airport... If you can catch a ride to one of the international hubs that has direct flights to Tokyo (DC, Houston, Denver, Chicago, New York, LA, etc.), your airfare will be a bit lower in most cases.
Whatever you choose to do, just make sure you sign up for your airline's frequent flyer program. Yeah, you get a bit of spam or junk mail, but it's free, and you never know when you'll take another trip. Those miles to Tokyo add up really quickly. I've gotten free upgrades to first class before, and, trust me, it makes a HUGE difference.
I've had the convenience of having my brother-in-law's parents working for the US military in Japan, which means I've had a place to stay every time I've gone there. But, they moved back to the US, so now I don't have a place to stay (one of the reasons why I'm hesitant to take trips to Japan now). Anyway, what this basically means is that I have absolutely no experience with finding places to stay while visiting Japan.
But, I can say that it's quite nice to stay in someone's home, rather than a hotel room... I never had to worry about meals... Breakfast, lunch, and dinner were always available for free, if I wanted to eat at the house. The amount you spend on food in Japan can get out of control really easily, so having that option saved me tons of money.
If you're feeling bold, you can always try Couchsurfing. Personally, that concept scares me, but I guess it's not too much different from staying in a hostel room with a bunch of other strangers.
Try to stay in an area that's close to where you plan to be most of the time. The house I was staying in was in Kanagawa, so I had to ride the train for about an hour just to get to Tokyo. Also, keep your train fares in mind because that can also get out of control really easily, especially if you start leaving the Tokyo area. I was spending well over 1000 yen each day just on train fares, but 800 of that was just because of that round trip between Kanagawa and Tokyo.
I've been to Fuji TV's Odaiba Gasshuukoku many times during my trips to Japan, and if you're a fan of any Fuji TV programming at all, it's a great event to go to, regardless of Idoling!!!'s activities. Plus, there's lots of shopping and other things to do in Odaiba. Anyway, there are basically two stages where live performances take place during Gasshuukoku: a small stage on the ground level outside the Fuji TV building and a big stage in another area down the street. This other area with the big stage gets a different name each year, but one thing never changes: you need to buy a Gasshuukoku pass to access this area. Attractions outside the Fuji TV building, on the other hand, are free and open to the public. Luckily, most of Idoling!!!'s daily mini-lives are on the small stage at the Fuji TV building.
I've never been to one of Idoling!!!'s lives on the big stage, but I've been to a bunch on the small stage. When they post the schedule for the Idoling!!! mini-lives on the Idoling!!! web site, they'll put up times to get "tickets". The tickets are free, and getting one lets you stand in the fenced area in front of the stage, but they're pretty useless unless you get there extra early and get first or second row. Why? Let me try to draw it out.
----------
| Stage |
----------
------------------
| Fenced |
| Area |
Hopefully you can see from this poor diagram that people without tickets can stand just to the left or right of the fenced-in area to get just as close as those who had tickets. Also, since it's outdoors, you can also go up a nearby staircase and take sneaky pictures, if you'd like.
So, it's not really worth it to try to get to Odaiba early for those tickets, but there is one other reason you might want to go early... They normally sell the Natsu Yasuming!!! DVDs early in the morning. These used to come with handshake tickets, but I don't know if they do that anymore. I'm not sure because I haven't actually bought any since 2010.
Unfortunately, I've never been to a Tokyo Idol Festival because they always announced them after I already planned out my trips and bought plane tickets. They finally announce it nice and early this year, which is the year I probably can't go.
But, from what I understand, it works somewhat similar to Gasshuukoku in that there are many events spread out around Odaiba, but obviously, they're all centered around idols instead of Fuji TV programming. But, I don't think there are any open areas, so a pass is required to access all of the TIF events. You can choose to buy a one or two day pass, and that will grant you access to any TIF event you wish to attend within that timeframe. But, again, this is just my understanding of how it works based on the materials I've read. Cicily has actually been to one before, so maybe he can tell you how it really works.