Wow - lots of opinions and options, I'll try to add some of mine along with relevant examples (I also have two kids, 1.5 and 3.5 yrs, and like you, much of my photography revolves around them).
If you want to shoot family shots indoors, a FF camera is what works best - on my 7D, I don't like to go over ISO 800, which just doesn't cut it indoors. On my 5DII, ISO 3200 is fine.
So, 5DII + 24-105mm for general use is optimal. Might want a Speedlite flash for extra light indoors (bounced off the ceiling, the light is soft). In a situation where you have control over the background, it does fine for portraits and can produce some nice results, for example:
EOS 5D Mark II, EF 24-105mm f/4L
IS USM @ 105mm, 1/60 s, f/4, ISO 400, 430EX II
The 70-200mm f/2.8L IS II is a wonderful lens, the images you get make it worth the cost and weight. I'll confirm your statement about the focal length being awkward on a 1.6x body - I found that to be the case personally. I love it on the 5DII. It does very well for outdoor portraits, where f/2.8 is enough to provide good background blur. An example from the combo:
EOS 5D Mark II, EF 70-200mm f/2.8L
IS II USM @ 70mm, 1/250 s, f/2.8, ISO 100
In most cases, f/2.8 on FF works very well for portraits. If you have a FF camera, I honestly think you'd be fine with the Canon 85mm f/1.8. It focuses very fast and accurately (AF accuracy is reportedly an issue with many Sigma lenses), and is very sharp. I do have the 85mm f/1.2L II, and it's wonderful for portraits...but, I usually use it in the f/1.8-2.2 range, else the DoF is just too thin. Some examples below, note how thin the DoF is, even at f/1.8 (often only one eye is in focus)...
EOS 5D Mark II, EF 85mm f/1.2L
II USM @ 1/60 s, f/1.8, ISO 400
EOS 5D Mark II, EF 85mm f/1.2L
II USM @ 1/160 s, f/1.8, ISO 100
Side note on the 85mm primes - if you plan to use these for outdoor portraits, get a 3-stop ND filter for the lens you buy. At f/1.8 on a sunny day, even 1/8000 s at ISO 50 can overexpose. I recommend 3 stops for the most flexibility - 2 stops might not be quite enough, and if 3 stops is alittle too much, you can always bump the ISO to 200 or 400 and that won't impact IQ. I use (and recommend) B+W filters.
The 135mm f/2L is also excellent - great for tight portraits, but as your kids start participating in sports/etc., it's also an excellent lens for that (kids sports are usually in poorly-lit venues, as are dance rectitals, school plays, etc.). Here's an action example with the 135L:
EOS 5D Mark II, EF 135mm f/2L
USM, 1/1600 s, f/2.2, ISO 100
I also really like the 35mm f/1.4L - for exactly the situations you describe, indoor activities with ambient (i.e. poor) light. It's also great as a nighttime walkaround lens, and can be fun to use for portraits on occasion. Here's an example:
EOS 5D Mark II, EF 35mm f/1.4L
USM, 1/30 s, f/1.4, ISO 100
So, my recommendation would be:Starters
EF 24-105mm f/4L IS
EF 70-200mm f/2.8L IS II
Speedlite 430EX II
EF 85mm f/1.8 (unless you really
want the wider aperture - I suspect you won't use it much)Later
EF 35mm f/1.4L
EF 135m f/2L
EF 1.4x III Extender (or the cheaper MkII), if you find 200mm isn't long enough sometimes
IMO, the 35L and 135L are likely to be updated reasonably soon. The 35L could use weather sealing and sharper corners, and the 70-200mm II is actaully sharper than the 135mm f/2L (not a knock on the prime, the zoom is truly excellent, but a prime should
be sharper, and I bet Canon will address that soon).
I don't really use the 100-400mm on my 5DII. I find that if I need more than 200mm (FF equivalent), I usually need way
more, so the 100-400mm on the 7D is the choice for distance shots (birds/wildlife), with 640mm FF equivalent at the long end. For occasional use, you can consider supplementing the 70-200 II with a 1.4x extender (giving you a 98-280mm f/4 lens with very good IQ). Personally, for the uses you indicate I find that 200mm is fine almost all of the time.
As others have mentioned, get a decent tripod - personally, I recommend Manfrotto as the best compromise between quality (very good) and cost (expensive, but not a Gitzo). If you get a cheap tripod, you won't use it because it's a pain to use and doesn't provide enough stability. My Manfrotto 190CXPRO4 and 468MGRC2 ballhead (498RC2 would be fine) supports a gripped body and 70-200 II or 100-400mm with no issues. If you don't plan to hike with the tripod, you can get aluminum legs (cheaper but heavier), but do keep in mind that if it's too heavy you won't bring it at all, and then it's just a waste of money.
Also, as you and others mention, the kit is pretty substantial and heavy. So, keep in mind the title of your post, and don't forget a good camera bag - or more than one. For the kit I suggest above, a Lowepro Flipside 300 would be a good choice, IMO. I also like the Lowepro Toploader Pro 65 AW for just camera and standard lens, 75 AW for camera and white telezoom, often with a Lens Case 1W attached to the side for a second lens.
Finally, I'll second what bycostello said - sometimes, you just can't take a dSLR and set of lenses with you. For those times, I use a PowerShot S95 - big sensor (for a P&S) means relatively low noise, and it shoots RAW for maximum post-porcessing flexibility.
Hope some of the above helps...and, good luck with your decisions!