Just like the description says, leveling up in the Dog Handler Skill is based on your time interacting with Dogs. It helps being at an event where the event is a Dog Event. Dogs all act different, they will pay attention to their owner more than someone who doesn't own them. Dogs will interact with someone who has a higher dog handler skill than others. Petting, talking and feeding dogs treats all help level up fast, but it's been said that the best way to increase the Dog Handler skill is to send your dog to someone so that the person who the dog goes to will hopefully pet it and interact with it. Dog Commands are not like emotes, you need to follow each 1 with the Enter key, and only use 1 every 2 seconds, anything faster and the dog tends to not pay attention. Here are the Dog Commands :

back
bad
ball
bark
beg
come
die
down
fetch
fight
get
go
good
heel
here
jump
kill
lie
nice
paw
play
run
shutup
sicem
sit
sleep
speak
stay
stupid
up
wag
walk

It's been said that some of the (not so nice) commands have an opposite effect on your skill, but that's only a rumor. I find that using 1 of each command works fine for me.

I experienced it myself. Continuously told my dog to die and I lost my casual Dog Handler status very quickly. I believe 'sicem' will also cause this effect. Positive commands with a nice 'good' in between seems to work just great.

Training the dog is not just a matter of throwing a bunch of commands from a list at the dog. Many of the commands only make sense based on the current dog position.

For example:

A dog must "lie" before it can sleep.
A dog cannot "jump" when it is lying on the ground or sitting - however it can jump when it is beg/paw/shake.ing

Petting your dog is a positive reward for doing what you asked - if you pet the dog first, it will just ignore you and wander off. Pet the dog for a job well done. Say "good" when the dog follows your lead as the alpha dog in the dog pack. Your dog wants you to be in charge.