Is it necessary to find the distribution of the data before fitting into a model?
No; it's neither necessary (since the unconditional distribution of data doesn't relate to the assumptions for the kind of models you mention) nor possible (you can't identify what distribution data come from, though you might be able to rule some out).
Is there a criteria for selecting a model based on distribution?
The distributional assumptions, if any, apply only to the responses (not any of the predictors). If you want to use (multivariate) linear regression or nonlinear regression, there's an assumption about the conditional distribution - and even then an assumption about the functional form of the conditional distribution really only when constructing confidence intervals or doing hypothesis tests. And even then, in large samples only the prediction interval would be particularly sensitive to moderate deviations from the assumption. (Assumptions like homoskedasticity and independence matter more than the distributional form, however.)
(If you were doing something like a factor analysis, you could have an unconditional-distribution assumption there.)
Should the number of variables in the model equal to the number of columns of data?
I don't think this is necessary. It depends on your model.
If so then it will not be an AR model, I think.
I don't see how that relates to the previous issue. The time series aspect is new with your edit, previous you looked to be asking about linear and nonlinear regression.
If your problem is a time series one, your tags should include 'time-series'.
I should then go for higher dimensional polynomials where each variable will represent a feature vector.
I don't even understand what you're saying now. This is because you didn't explain your situation or underlying conceptual models nearly clearly enough. Most people here will not know what you're doing. I certainly didn't even realize from your original question you wanted to ask about time series models. I have no idea what you mean by 'feature vector' in this context.
You jump from mentioning a few of your variables to 'can anyone explain how to do this'... without really explaining what you are trying to achieve. You seem to think we can read your mind or that we know what you know. We really don't.
Or are there other linear or nonlinear regression models?
Certainly there are other models than AR or models based on polynomials.
I am particularly interested in NARMAX modelling but do not understand the complexities.
I'm unfamiliar with the various issues with fitting nonlinear time series models with exogenous inputs like NARMAX (Wikipedia on NARX for anyone else unfamiliar); if you were considering this kind of analysis, your tags should at least include 'time-series'.
After fitting into the model,how do I test the "goodness of fit" ?
Why test, specifically, rather than assess?
