That battery should be fine since it's the same number of cells and same discharge rating. The extra mAh should get you a bit more flight time, but it's also going to add weight, so it's not going to be as much as you think. Also, with the added weight, the copter may be slower/ less agile, but that's probably a good thing if you're just starting out - and I doubt it will even be noticeable.
I think you have the wrong connectors.
The battery uses 4mm bullet connectors according to the comments on that page (and that size normally would use 16awg wire/ 4mm connectors. The motors are almost for sure going to need 3.5mm bullet connectors. They may come with connectors already on them, but the ESC's most likely won't.
From your setup, I'm seeing:
Battery - 16awg wire/ 4mm bullet connectors (one female, one male) *if you go with the original battery, it has deans or T connectors. You could always cut those off and replace with bullet if you prefer.
ESC to Power - 12awg wire (x2), no connectors installed - use 3.5mm bullet connectors
ESC to Motor - 12awg wire (x3), no connectors installed. - use 3.5 bullet connectors
Motor to ESC - 12awg wire (x3), many motors come with all male 3.5mm bullet connectors installed, but if not, you can do them yourself.
You're going to need:
1.
4mm bullet connectors with protectors (the protectors with male/female).
http://www.hobbyking.com/hobbyking/store/__9283__HXT_4mm_Gold_Connector_w_Protector_10pcs_set_.html
This is only for the battery to power spider connection, but you want a whole pack to be safe in case you mess up. Also, the "protectors" need to go on before you solder on the connectors. You can slide them up the wire, then snap them back into place once the connectors are on. Alternatively, you can save yourself a lot of time and soldering by going with one of these:
http://www.hobbyking.com/hobbyking/...ullet_Multistar_ESC_Power_Breakout_Cable.html
If you go that route, you'll need to plan ahead to make sure all your wires will have enough length so you don't have to extend them. This could be tricky since batteries and ESCs don't always list the length of their wires in their specs.
OR you could do something like this:
http://www.hobbyking.com/hobbyking/...King_Quadcopter_Power_Distribution_Board.html
If you do that, you'll still need the 4mm connectors because you'll have to chop off the deans connector and replace it.
2.
3.5mm bullet connectors with protectors (or without protectors). You'll be using these for ESC to power and ESC to motor connections. You'll most likely use heat shrink rather than "protectors", so you can get them without protectors instead to save a few cents. You could use the protectors though - it's up to you. You'll just have to make sure to get the right male/female combinations for your setup.
3.
12 and 16 awg wire. I like the silicon wire from either HobbyKing or RC Timer. You can probably get away with a meter of each size in any color, but you may want to get a meter of red and a meter of black in each size so you don't confuse power connections. You can also buy spools of silicon wire from other places.
4.
Heat Shrink - I think you'll need 4mm for 3.5 connectors. I'm not 100% sure though - I buy a lot of different sizes and just eyeball them to see which one will fit over the connector. 4 should slide right over a 3.5 though and the wire is much smaller. You'll need a heat gun to get this to shrink. A lighter also works. Heat guns are $10 at Harbor Freight.
