Is it okay to modify the LaTeX template for article submission with additional packages?

Unless explicitly forbidden, adding packages is usually ok and fully expected - e.g. in order to add support for listings, tables, symbols, encodings, operational utilities (extended macro/logic support), or specific features thereof. This should, however, never mean a general change in the appearance of the paper. Therefore, packages that globally alter fonts or font sizes are usually not allowed.

As for using a separate .bib file, you can usually work with that and (if explicitly requested by the guidelines - this varies a lot by publisher) copy the formatted information into the .tex file as late as for the camera-ready version (when nothing changes any more).

I usually publish in IOP journals and I can confirm O. R. Mapper's remarks.

As an example, these are the packages included in a recent paper of mine:

\documentclass[10pt,a4paper]{iopart}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{iopams}
\usepackage{amsopn}
\usepackage{graphicx}
\usepackage{cite}
\usepackage{siunitx}
\usepackage{tikz}
\usepackage{array}
\usepackage{subfigure}
\usepackage{longtable}
\usepackage{listings}

Notice that for what concerns mathematics, there are a few idiosyncrasies related to the iopart class: e.g., this class is incompatible with the amsmath package, and it is suggested to use the eqnarray environment, which is otherwise deprecated. I thus suggest a careful reading of the IOP LaTeX guidelines, which can be found here.

A few notes on the example above:

• The package iopams is an extension for amssymb to provide predefined names for bold greek letters (thanks to Barbara Beeton for pointing out this and Canageek for reporting it here).
• The package subfigure is obsolete but the the newer subfig package seems to be incompatible with iopart because it includes the caption package.

As someone who moonlights as a type-setter for a mathematical journal let me tell you something about bibliography.

Bibliography styles are different from one journal to another. There is nothing more terrible than having to reformat 60 entries by hand, and there aren't many things worse than having to reconstruct the .bib file in order to use the journal bibliography style more easily.

Sending the .bib file to the publisher is not a bad idea. If they can handle LaTeX, they can handle bibtex as well. I never worked with biblatex personally, so I can't quite tell you about that. But I do recommend that you submit you .bib file separately, and save the hassle to the type-setter.

Of course, this assumes that your .bib file wasn't hand-made. When you add an entry by hand, it has a high percentage of chance not being up to standards. Always use MathSciNet, if not then ZBMath, and if not, then Google Scholar as a last resort.