For example, there's like a million files stored on a website (publicly, anyone with a link can get it), and I wrote a program to retrieve it one by one. Since obviously my program pings the server constantly for the files and can potentially crash the server, does this consider a small, tiny form of DDoS?
I'd say no. A DDoS is a distributed denial-of-service attack. Writing a program to sequentially download files stored on a server certainly isn't distributed as long as you plan to run the program on a single computer. It's not an attack if a) you don't intend to crash the server and b) there's no reason to believe running your program will do so.
If the goal of your program is to crash the server (and your program meaningfully restricts bandwidth) then running the program could be considered a DoS (plain denial-of-service attack). But even if you don't intend to crash the server but there's a good chance your program could do so, then your actions could be considered an attack.
No. This is actually a different kind of attack called Bandwidth Starvation, sometimes called bandwidth raep. If memory serves me correctly, bandwidth starvation's goal is to suck down all of the bandwith that a server can use by downloading the large file. Although it is still a form of denial of service, it is not closely related to the packet spamming, UDP spoofing, and power consuming attacks that distributed denial of service attack is normally associated with.
It is unfortunate that I must cite such an unreliable source for security, but this is the best article I could think of to explain the difference between the two. Although they are similar, they use different tactics and result in different outcome.
One thing to consider, bandwidth starvation is an obsolete attack, coming from the same age as IMAP ping attacks. Most servers have enough bandwidth to render this kind of attack useless. It would take the repeated download of a file that is several gigabytes in size across several hosts (most likely attack servers or computers within a botnet), to preform this attack.