Yes. Most people back in 2001 were not using fibre. Dial-up, DSL, etc are all much higher latency. In Dial-up you are easily talking at least 100ms slower, if not a lot more.
You bet. Mangonel micro, archer kiting, cavalry bluff charges, deer pushing etc are all things you regularly see today that wouldn't have been feasible in the old versions. (in online play)
Yes: dialup modem latency was on the order of 100-250ms, DSL 10-70ms, and cable modems 5-40ms.
Sitting on my couch using Wi-Fi over cable, I have 18ms to Google.com – call it a full order of magnitude better than a disk up user despite Wi-Fi adding 7ms.
Last mile latency has been reduced a lot as people moved to broadband, but also on longer distance routing where there are more paths from A to B than twenty years ago, and some of them are physically shorter. Higher bandwidth interconnects also mean less time in transit; it's not a lot, but it is some.
Also, some early broadband networks backhauled too much traffic to central locations. I recall commonly seeing traceroutes from my home in orange county, ca going to kansas, and then coming back to servers in los angeles. I still see some traffic that leaves my metro area only to come back, but it's less frequent, and it's usually only going one or two states instead of halfway across the country.