One observation: powering the servos from the +5V Arduino rail could lead to damage to the chips on the Arduino - its a risk I certainly wouldn't take, unless I had a spare board and didn't care about the expense. Servos and motors produce nasty voltage and current spikes and generally you never share motor power with logic if you want reliability.
Secondly you really should get a multimeter and measure the current consumption of one of your servos (while continuously moving) to get a better estimate of the power supply / batteries you will need. For now I'd suggest 4xAA battery pack for the servos only, share a common ground and run Arduino from USB. At a later point you'll need to work out how to power the Arduino from a separate rail to the servos (whether via a LDO voltage regulator and / or filter circuit, or DC->DC converter, or separate battery).
If I connect 6 servos in parallel and each need 1A, would I need to produce 6A from the power source or would they share it like in voltage and 1A is all that is needed from the power source?
If they are active at the same time, 6A, if only one is moving at once, perhaps 2 or 3A (they take some current when parked).
If I need to produce 6A, is that dangerous when the voltage is somewhere in the region of 9v-12v? I only ask as this link and one on wikipedia claims that above 2,000mA (2A) can produce some quite nasty results, but I'm not sure if it's only when a certain voltage is present (my guess is that is the case, but wanted to be sure before frying myself)
Unless you put wires on your tongue, 9 or 12V is perfectly safe. Large currents can lead to heating issues, but not shocks.
If I connect 4 AA batteries in series, my understanding is this will increase the voltage but the current will be the same as 1 AA battery. So if 1 AA battery is 1.5v and 1800mA/hr, it would be 6volts and 1.8A maximum for 1 hour. If you do it in parallel, it increases the max current but not the voltage. So in this case it would be 1.5v and 7.2A maximum for 1 hour
Firstly units. Current is measured in amps, charge (capacity) is measured in coulombs or ampere-hours (Ah). amps/hour is a rate of change of current, completely wrong here.
A measure of capacity (1800mAh, not mA/h) says nothing about how much current can safely flow. You need to know the maximum discharge rate for that - anyway lets assume these hypothetical batteries have a maximum discharge rate of 1.8A, then yes in series the rate is unchanged, in parallel it adds.
Different battery chemistries have different properties (for instance most NiMH rechargeables would only be happy discharging at the hour rate or so, whereas a LiPoly battery might be rated to fully discharge in 120 seconds!).
Lastly if you buy some 2Ah batteries, assume their capacity will be down to 1Ah before too long - in the real world rechargeable batteries perform quite poorly without careful management and manufacturers bend the truth to breaking point. And over-discharging many kinds of rechargeables completely knackers(*) them, especially when in series as one cell may get reverse-charged.
(*) capacity reduced by large factors.