Dell Inspiron = Good
Processor = I'm sure it's adequate unless you have heavy needs. There are many types of i3 but chances are it's faster than anything you have now.
Windows = Which one? Win8 or Win7? There is a world of difference between these two. Win8 looks like nothing Microsoft has ever put out and there is a steep learning curve associated with it. There is a way to make Win8 look like a Win 7 system. I have a Win 8 laptop and my son should be able to figure it out since he's the main user. I would like Win8 better on a tablet or iphone or something. I'm learning it so it's not impossible.
Memory = 6 Gb is very good
Hard drive = If you aren't storing a lot of files and data on the laptop, the size should be more than fine. I don't store a lot on my laptop but my desktop is full of very large uncompressed music files and I got a 1 terabyte external drive that I quickly filled up. This is my newest laptop and I have a 500 Gb hard drive which is plenty for programs and stuff.
Battery life - 4 hours should be good unless you carry your laptop around on the road a lot without places to plug into.
Screen = 15" is a good size. I have bad eyes and don't have much problem reading. There is a magnifier app that helps if something is small and many programs allow you to zoom. Resolution is as important as size, but my experience with the Inspiron computers is A-ok
Wireless mouse and keyboard is great.
From what you say you need, this is a good choice.
It's a crapshoot whether or not your games are backward compatible.