If that router is one with several (probably four) ports on it, connected with "Cat 5" cables, then it will work just fine with your new PC as-is.
Further, if you have space you could keep the old PC in operation alongside the new. You could then "network" them (instructions available on here) to copy stuff between them.
Software is very much an individual choice - people like what they use, and despise what they don't. Hence some use the proper name for stuff, and others use the mocking name - that tells you which they use.
I happen to prefer Thunderbird for mail, because I don't use webmail and I have a lot of separate domains for different purposes, and it handles those well. Outlook puts everything into one stonking great file (or did, last time I used it) so one glitch lost ALL your address book, ALL your mail history, and more besides. Once bitten... Backup is limited help, since the crash will lose the most recent stuff that hasn't been backed up yet and that you haven't dealt with.
IE and Firefox play catch-me-if-you-can with each other. I didn't like the way IE told me how I was to use my PC (again, years ago) so I switched to Mozilla which became Firefox. It's served me very well so I don't bother to change. I keep IE on the machine for those very few websites that use Microsoft's non-standard HTML and therefore don't work with Firefox or any other browser.
Windows 7 is, in my experience, as good as Vista is bad. It was good enough to get me off the Linux hook - the Linux machine is now out in the workshop, not connected up since the move.