Lots of options, depending on budget. I can say that I know where you're coming from - I have daughters nearly 5 and nearly 3, and the 5DII's AF struggled to keep up (and usually failed). The 7D AF did much better, but the ISO noise limited me to 1600 or so (that's a subjective call), meaning useless for most indoor ambient shooting. I did use the 7D for birds in good light, where the 1.6x helps get more pixels on target.
The 1DIV is an excellent camera for action/wildlife, great AF, fast frame rate, and the 1.3x crop give more reach than FF. But, not so good for landscape (your wide angle 35L becomes a 46mm 'normal' lens, for example). Also, your DoF gets deeper for portraits - the 135/2 becomes a 175mm f/2.6 in terms of framing and DoF for an equivalent shot to FF.
When you say 'Mark III' do you mean 1D or 1Ds? The 1DIII will have the same crop factor issues as the 1DIV for landscape/portraits; the 1DIV has better AF than the 1DIII, but the III is still miles ahead of the 5DII and will meet your needs fine. If you mean 1DsIII, AF is the same as the 1DIV, but you can't push the ISO up very high.
The 5DIII is a huge step up from the 5DII in every way except IQ (and the 5DII is already great there). It has the 1D X AF sensor and 6 fps, truly a great all-around camera.
The 1D X just can't be beat - top IQ, top AF, great build and ergonomics (for some - I used gripped bodies before getting the 1D X, if you don't it will be big and heavy by comparison to a 5DII). Although the 5DIII has the 1D X AF sensor, the 1D X has a much better metering sensor, and data from that are used for face-tracking AF - I can tell you that it locks onto my kids' faces and keeps a lock as they run all over.
The only 'downside' to the 1D X and 5DIII is when it comes to birds, the FF sensor means less apparent reach. But if you're shooting them with a 5DII, it'll be no worse. Coming from a 7D + 100-400mm, my 'solution' to the loss of the 1.6x was to buy a 600/4 lens - but at $13K that's not an option for many people.
For your family and landscape shots, I'd stick with FF. But there are a couple of possibilities there: 1D X, or a 5DIII and $3200 left - that could buy a couple of nice lenses including a 70-300L or 100-400L for birds, 24mm TS-E for landscapes, or even a 7D and 400mm f/5.6 which would be a great BIF setup.
Hope that helps...