Your old router is pretty bad for wireless and it's probably very lack on security as well given Dlink
didn't updated it in three and half years which is same as saying since dinosaurs roamed the earth, in a security world.
http://www.smallnetbuilder.com/wireless/wireless-reviews/31545-d-link-dir-657-hd-media-router-1000-reviewed?showall=&start=2
There no benefits (in your case) to switch from cat5e to cat6, unless you plan to run 10Gig speeds at home
How fast is your internet if you test from wired computer. I prefer this test:
https://www.dslreports.com/speedtest
Your smartphone test could be not accurate if you used your Cell provided network instead of wifi to test it, but even if you did test with wifi on, it still very much will vary on crapton of variables and in many cases you won't get on the wifi top speed avaialbe by your ISP.