Indeed. Cut 1 wire of your battery at a time, don't cut both + and - wire at the same time, cause your scissors will create a short.
When soldering, you should see the liquid solder flow around the pin and sinking in the hole. If it's then shiny when solidified, it should be a good solderjoint. Heat the pin (at the short end), not the solder. Then hold the solderwire against the hot pin to let it melt and flow around the pin and into the hole and on the solderpad.
Lead Free solder just requires a higher temperature to melt. 60/40 tin/lead melts around 200-degrees(f). Depending on the alloys used to replace the lead, the melting point will be as high as 250- to 300-degrees(f). This can lead to "cold" solder joints.
"Rosin core" and "Lead Free" are not related issues.
The rosin core reduces the oxidation of the solder (oxidation makes poor joints). It also makes the solder "flow" better into the joint. If the work "puffs" smoke while soldering, then you have a rosin core solder and the puffs of smoke are the rosin boiling off. What u4eake said - if the joint is shiny when it cools, then you likely have a good solder joint.
Radio Shack has a beginners soldering kit.
That battery will do just fine and will provide good flight time. The connector that comes on the battery is matched to the current it can deliver which is probably more than you need in this situation. Replacing the connector on the battery is one way to get it to work (just don't use it where more current is needed than the connector can deliver) or you could solder up an adapter to go between the two connectors so you can use the battery in higher current applications down the road.
thanks for the advice, and I did one at a time. Seems to have worked out. I really like the rosin as it seem to work easier.