Regarding 1. (I cannot answer 2., on the one hand, for lack of expertise in this field, and on the other hand it will be difficult to answer without detailed information; perhaps you can ask somebody senior you know for advice?)

I do not know the general MO-view and I am not around since the start of this site, but it is my understanding that, yes, numerics is rare on MO put merely due to the fact that relatively few regular users of MO have this as their center of interest and thus few question/answers come up and thus few people with these interest become regular users and thus...a vicious circle.

But, numerics is by no means the only field which is not present on MO relative to its general relevance. I believe there are two phenomena:

a. as described above, if many people in a given subfield are one MO, then even more arrive. (I do not know the initial distribution of the user base well; but I beleive that simply due to the respective interests of the people who started the site there is some bias that propagated. Not because somebody wants it to be like this, but just because it happended like this and essentially cannot be changed. In an alternate world where some people in numerics of PDEs rather than algebraic geometers or topologists had started the site it might look very different.)

b. I believe some fields are better suited for the format. In fields with a lot of theory there are more questions somebody might have that somebody else can (easily) answer. However, in fact, based on my vague understanding, I always imagined numerics could be a good fit for this type of questions as it is (in certain aspects) a bit vague and thus experience (that can be communicated) is important.

Regarding numerics in general my personal view. It is not my field, but I am aware of some things I find very interesting (mathematically) and I am convinced there are many more (which I simply have never heard of). Certainly not a 'border group.' Is the thing with the quantum computers a trick-question?

As I write this, there are 190 questions tagged as numerical analysis, and many of them have received insightful answers. Questions about numerics are welcome on MathOverflow, as long as they are well-defined mathematical questions and at an appropriate level. The first question you are asking does not seem to fit these criteria, since it seems to be an opinion survey. I can't speak for others, but I doubt there will be a "general MO-view" of numerics, since the users form a rather diverse group.

I'm not a computer scientist, but as far as I know, the only known advantages of quantum computers are derived from database searching and Fourier transforms in black-box abelian groups. In particular, I don't know of any evidence that quantum computers will make the typical numerical problem any easier to solve.

Regarding the second question, what do you mean when you say that your numerical method is "more accurate" than existing methods? My wild guess is that you have in mind a specific class of problems where you can bound the error in terms of mesh size, and for that class, your method has better error asymptotics. Is that an accurate interpretation? If so, your results are probably publishable. I see that you asked a question about this last month. Have you followed up on the reference in Nilima's answer?

I will try to give a very general answer to your 2nd (now only) question.

First, you should talk to somebody more senior who knows the field and the precise situation. In view of your age (as given in your user profile) and as you say it was supposed to be your first paper, I assume you have some kind of advisor or senior collegue with whom you are working with. If she/he thinks it is a good idea to publish it then try it, if not then not. The idea that somebody else will improve your work seems unlikely to me; except for a 'local' person, say the person advising you, somebody else in the 'group', somebody else this person knows. To find some good collaborator you do not already know (at least indirectly) for an ongoing project on-the-fly seems very unlikely. So, again, you should talk to somebody more senior who knows the field and the precise situation.

Finally, while you retracted it, let me add: you really should not worry about generalities some other mathematicians say on another field. For essentially any pair of maths subfields (A,B) you will find somebody that will explain you why A is way more important than B (and yes often based on highly incomplete information). If an expert in your field tells you that your precise research direction does not seem so fruitful, you should pay attention. But which subfield is 'the best,' is a question of the type which [insert random sport]-team is the best; discuss it if you enjoy it, if not, ignore the discussion.