ENTP here. I'll do my best to answer your questions so I don't have to sleep.
What does the stereotype "Ne-doms are always coming up with new possibilities and ideas from everything they see" mean?
It's more like if I see something out of the ordinary, my mind will spin on it. Not EVERYTHING I see. But it's like walking into a run-down room and knowing I could turn it into the best room in the house. Having vision. Imagining what colors and window frames and furniture would make this the best room ever. A lot of these ideas aren't even feasible, or are something I don't care to invest time and money in. But it's just a matter of being attuned to the what"s around you and inspiration sparking a (potential) vision based on it.
How's the reality?
I'm not sure if you mean this as a follow up question, or my relationship to reality. All I can say on the latter front is that I actually have a very strong sense of reality. I test as a Sensor for this reason--I just don't have too much use for people who live in a fantasy about the absolute nature of reality. What Ne does for me is perceive it, and then expand off it. So I'm visionary. I see what is, and I see what more it could be. This can be something simple like seeing a coat-hanger and getting the idea for a creative project, to seeing my country and having a vision for how to improve social equality.
How often do you have new ideas?
It's not like I can quantify that...every five minutes, or 3 times daily. Nope. Instead, it's more like I move through life guided by larger ideas. I could study in X country. I could move to Z land. I could make a movie about this, let's get funding! Most of my ideas aren't really new, but instead just an inspiration. It's a thought process, so it happens minute to minute, but most of the ideas aren't as big as making a movie. "I could look up XYandZ from my childhood", "Hey I wonder how they make taffy? I should look it up!" "That skyscraper looks like a penis". "I should write a story about this". Those are the smaller ideas.
What kind of ideas do you have, ideas for a new invention?
I think I referred to some above. I don't really have much patience for inventing things per se. It's more like, Hey I can use these old egg cartons as planters for my beans. Hey, that pretty stone would look great on a necklace. Aww, look at that dead squirrel
*imagines bereaved squirrel family and funeral*. It's typically called "creativity" by those who don't understand the functions.
What defines an idea?
Something abstract, still in your head, that connects irl things in ways that haven't been thought of yet. Idk. Don't listen to that.
Do you have so many ideas on a daily basis that you could write hundreds of books because you have so many ideas for their stories?
I personally am something of a story teller, so yes, I invent all these epic stories inside my own mind. Hundreds and hundreds exist. Some are just scenes (eg that dead squirrel), some are entire overarching ideas with nothing to fill in (eg, What if African nations had developed technology before the Europeans? I should make a graphic novel about that world), and some are entire story lines (I wrote a 30 page summary of a guy who takes over a nation last year, not Donald Trump incidentally). I could seriously do this for a living, but I'm fated to die without sharing any.
How does your dominant Ne affect your daily life?
Too broad for me to know where to even start...
I live in a world where random things that catch my interest put me into a cinematic fantasy, where being creative and imaginative come second-nature, where more and more tabs get opened and I click between them mid-thought, where people think I'm psychic, and I could give two shits about tidying my personal spaces because my inspirations are more awesome.
The most annoying thing is Si-inf, and how I forget daily detail stuff. Like in one ear out the other. Except I misplace things every 20 seconds, too. Like Dory, I have limited short term memory, so that while the class describes what they ate for lunch to each other, I'm like, Wait, I ate lunch?? I have bruises and no idea how I got them. RABIES. Because I'm following what's really important, which is my vision...not my lunch and not my body and not what so-and-so's age and name are.
Could you give real life examples? 
What specifically do you want to know??