How to increase the size of first character in a chapter (Drop-Caps)

These are called dropcaps or lettrine (from the French). The best package to use is lettrine which is available from ctan.

Use as:

\documentclass{article}
\usepackage{lettrine}
\begin{document}
    \lettrine{A}{gain find} more words her...
\end{document}

Here is the output:

They are very difficult to handle typographically, unless the whole page design has been developed with dropcaps incorporated. The modern trend is to use a sans serif font rather than a serified for the dropcap.

Can Microsoft do it? It can, but not recommended to be used for typesetting books.

1. (Dropped) initial (and some small caps following). See eg here: http://www.ctan.org/keyword/dropped

2. See eg http://tex.blogoverflow.com/2011/08/putting-colors-in-initials/

3. I don't know.

4. Probably yes.

If you're willing to use the Linux Libertine and Biolinum fonts, loading them with the command

\usepackage{libertine}

in the preamble, you can create drop-caps with the commands

\LlettrineS[<height in number of lines>]{<letter>} % w/ border
\LlettrineM[<height in number of lines>]{<letter>} % w/ mosaic

\renewcommand*\LlettrineDline{<height in number of lines>} % followed by
\LlettrineD{<letter>} % for display-style letters

There are also related commands if you want to use the Biolinum (sans-serif) font. Overall, the manual says that the production of drop caps ("lettrine"s) is still in development. At this time, this approach may be a bit experimental... Give it a chance and see if you like the results...