"which distro is the most stable / fastest I was going to try Centos"
All the droplet OS installed on Digital Ocean are from stable release channels. Picking which Linux to use is more a personal choice, but if you want to maximize resources on a small node some distros have a lighter footprint than others.
I use Ubuntu because I can find install packages for it for most stuff I want.
"if I select LAMP under applications during the creation of my droplet does that install all the web services - so no need to manually install them"
Yes, the LAMP option for droplets comes pre-installed with what you would get if you ran
apt-get install lamp-server^. It does look like DO has made a few tweaks to finish the install for you. So all you have to do is set the root password for MySQL.
Take a look at the other developer stacks available as well. There is the LEMP and MEAN stacks.
"I develop websites and play around with web scripts and I thought this would be a great way to pick up some linux skills"
Create a droplet, play with it, destroy it and repeat. I've tried a bunch of the pre-configured applications. Some of the ones I recommend try are
Docker and the