How long does clicker training take? I don't have that much time to spend training?
Then you're going to love this! Formal clicker-training sessions really should only last about 5 minutes (10 minutes tops). Less than that if you find that your puppy loses interest. Two or three mini sessions a day is ideal. That's it. That's all it takes.
Isn't my dog going to be dependent on treats this way? Am I going to have to carry treats around with me all the time?
No and No. Once your dog has solidly learned a behavior you can wean the treats. This is commonly done by asking for twofers then threefers (ie 2 sits instead of 1 gets a treat, then 3 sits, etc) or by asking for more than 1 behavior and then treating (like sit & down). Of course, it's a good idea to occasionally reward good behavior indefinitely (but it doesn't always have to be with a treat).
A behavior that is positively reinforced will increase in frequency!
You won't have to always carry treats. But, to tell you the truth, I usually do. I just never know when that little rascal is going to do something that I want to reward!