It's hard if you don't know what you are doing.
1) You don't solder naked 18650s, you spot-weld them. Soldering naked cells - that's what may cause explosion. On the + terminal you can easily slip a bit or drop a tad of solder under the Teflon ring and cause a dead short. Also overheating a "+" terminal may trip a protection valve and make your battery dead-dead(it's a "pressure valve", and even though it should recover in theory - it only happens 50% of time).
Spot welders are cheap nowadays. I bought myself a battery-powered one for ~$50, and just in the first week it paid itself off 5 times over.
Full charge is enough to repackage something significant, like a 48-cell battery pack.
2) If you don't want to buy a spot welder - buy solderable cells with nickel tabs.
This is the same 18650 Lithium Ion Cell that you know and love, but now includes pre-attached solder tabs! These round high capacity cells have been mainly
www.sparkfun.com
Which laptop do you have? If it's an older thinkpad - then you may be in luck. They have a recalibration tool.
I did it a couple of times on T540/W541, all it needs is carefully replace old cells with new ones, hook up power in the correct order from "-" to "+" one by one(so you won't trip the built-in protection), and do a full calibration cycle(it does a full discarge-charge routine and save the new capacity).
De-soldering also needs to be done in the correct order, from "+" to "-".