As far as pro/anti-drop and one word sentences: English is pro drop with regards to commands (in fact, it enforces* subject dropping in commands), but fairly strictly anti-drop otherwise. But you can still get a verb-only sentence if you make a command out of an intransitive verb (one that can't take an object), e.g. "Stop!"
Yes, that's true. In recent years, however, I have noticed a trend in informal writing to be pro-drop even at the start of normal paragraphs (which never really set right with me -- it just sounds "wrong"). To demonstrate, let's say I'm starting off a new paragraph about my visit to the store today. Here is a grammatically correct sentence:
"I went to the store to buy some milk today. ..."
In recent years, however, I have seen the pro-drop form as well:
"Went to the store to buy some milk today. ..."
Until a few years ago I would only see the latter style in special situations such as when writing a post card and the desire was come across as deliberately affectionate, informal, and brief. For example:
"Having a great time here in Phoenix! Found a great hotel. Rented a Corvette. Saw a HUGE cactus.
See you soon,
Joe."
But over the past few years I've noticed that style of writing in many Internet forum posts and emails as well. Perhaps it's just English "evolving", but it can lead to ambiguity depending on the context, such as across a paragraph break. For example:
"Selene and I were out running errands all week. ...
Went to the store to buy a new TV. ..."
So who went to the store to buy a new TV? Both Selene and I? Just me? Just Selene? Or my great-aunt Ethel? The second sentence does not have a subject and so technically it does not conform to English grammar rules. Perhaps the informal rule is that if no subject exists then the pronoun "I" is always assumed, but TBH it just sounds wrong (and choppy) to me to lead off a normal paragraph with a pro-drop. Maybe it's just a matter of what a person gets "used to" when learning the language.