Atom BASIC, anyone here used it or can emulate it?
There are several emulators, for example JSACORN whoch can do several Acorn models, including the Atom. It maps the special keys COPY to F12 and BREAK to TAB. All issued mentioned in the question can be tried and will work (AFAICT) work exactly as on a real Acorn.
..."Atomic Theory and Applications" (best name EVAR) ...
:)
For one, did the Atom have a "full screen" editor in the fashion of Commodore in that you could cursor-up onto a line, edit it, and press return?
No. At least not in the same sense as the Commodore
It's not clear in the book.
I have a hard time to find any hint for a full screen editor. For one, Section 1.7 - located on the same page 7 - describes the very basic BASIC editing known since the Dartmouth days:

So, no.
Except, there are cursor keys and a wondrous COPY keys ... and section 18.4 Screen Editing:

So, no (again), it's not a screen editor like on the commodores, but yes, it is a screen editor in the sense that one can move the cursor around and collect characters (and words). Moving has no influence on any program text, or line entered. A character is only added to the input line when it is copied from screen location via COPY at it's location. It's all about adding characters (or words) to the input buffer. Whenever return is pressed, that collected buffer gets taken as the line to be processed (added/replaced) within the editor text (program).
It's a bit like the EDIT command known from Microsoft BASIC 5 and above. But it could also be seen as a close take from the Apple II Monitor edit functions: Here free cursor movement around the screen is done using ESC+A, ESC+B, ESC+C, ESC+D for right/left/down/up (*2) while the right pointing arrow key has the same function as the Atom's COPY key.
With keeping this in mind its 'kind' of a screen editor.
On page 7 there are examples of typing in random text as lines. There's a mention on the next page about errors, but I can't tell if this is referring to errors caused by those lines or not?
Of course not - It would be a real bad manual showing examples that can not be repeated the very same way. That section (1.9) is simply to inform a new user about what error messages are - which he may already have encountered.
I suspect this is simply an example of basic typing, and that it would not allow you to store it as-is, but I cannot tell for sure.
Well, it does. One of the features about the Atom's BASIC, as of all other Acorn (at least 6502 and as I know) is in fact to allow to use the editor to store arbitrary text (*1).
Another... what exactly does OLD do?
OLD tries to restore an existing (program) text. It is in some way the direct help when typing NEW by accident - but as mentioned on Page 7 (real important page):

It's helpful to recover from a 'typing BREAK'. This might be a bit misleading as 'BREAK' is not something typed, but a dedicated BREAK key getting pressed. It's function is resetting the CPU.
Unlike with the Apple II where a warm reset keeps an existing memory content intact, the Atom clears existing memory as well - essentially like typing NEW
- destroing all refference to any edit data (aka 'the program'). Using OLD
will reinstate those with default values (*3).
For that matter, what does NEW do?
Same as NEW
on a Commodore: resetting the editor content to none.
The manual, page 7, says NEW "will clear the stored text so that a new piece of text can be typed in". Is "text" "the program" in this case?
Sure, if it was a program. The language reflects that Acorn sees its editor more universal as other BASICs.
And is "piece" "the entire thing"? And what is the purpose of OLD, which "can be typed after typing BREAK to retrieve the text previously in memory". That sounds like LIST
Or is BREAK something different too?
BREAK is a reset key and works much like resetting a Commodore (if a reset key is installed): Everything is returned to default, with the editor put in a NEW-state.
*1 - In fact that can be done with many other micro BASIC as well, including the mentioned Commodore BASIC. A feature used for example by some (simple) assemblers that use the BASIC editor for source handling. Only issue here is that MS-BASICs, like the PET-BASIC, 'crunch' lines entered, that is detect BASIC keywords and replace them by shortcuts (tokens). Using such keywords may screw the visualy, but then again, usually the assemblers are made to handle it anyway.
A notable exception would be Apples adaption of Microsoft BASIC aka Applesoft, as it's keywords include spaces when listed, so a line typed as
10IT'S CALLED PRINTING WHEN GRAPHICS ARE DONE BY A MACHINE
will be reproduced as
10 IT'S CALL ED PRINT ING WHEN GR APHICS ARE DONE BY A MACHINE
*2 - Note, this not being down/up/right/left :))
*3 - Would have been useful for a Commodore as well :))
NEW
clear the current program andOLD
retrieves a program if you cleared it but change your mind without putting anything else in memory. The keyboard has a COPY key so a two-cursor system for editing lines is a guess based on Acorn’s later machines, but I don’t know, or I’d post an answer. Can emulate in-browser at econet.network/atom but I’m on a phone right now so that’s all I can say about that.