Someone help me understand...

Then first of all, you're gonna need to get off your rhetorical questions and your high horse attitude.

Mages are ability casters like Xerath and Viktor . Incredibly stable to play, but many of them are both slow and immobile. You said Evelynn runs faster right? The only movement speed in Evelynn's kit is on her E - Whiplash. It only applies after she hits you. This means that if she's outrunning you to proc her charm, you're just playing a fucking slow character. I looked at your games and saw Ashe and Corki. Corki's almost a mage but both of them are really naturally slow - that's probably why you think Evelynn is fast.

Your 4 button in game places a ward down on the map. It clears up Fog of War so you can see further. Also, Evelynn bust come out of stealth to attack. In other words, if she wants to attack Jungle Monsters, she must be visible. To wards. So put wards on top of the Jungle camps she likes to take, you'll see where she is and where she plans to move next. If she's at Blue Buff and she moves straight into River coming right at you, run. Or at least back up.

If you don't know about wards, I assume you also don't know about Control Wards. They reveal invisible characters.

Take your minions and your enemy minions and draw an imaginary line between them. Shoving means pushing that line closer to the enemy, you do this by hitting the minions more than your opponent. Since you're a new player, this is the opposite of what you want to do in many cases. When you shove, when you hit the opponent's minions more than they hit yours, it means you're going to push the minions into enemy territory. It means you're putting yourself in enemy territory. And that means you're super easy to gank. It doesn't matter if it's Evelynn or not, any Jungler would see a shoved Ashe as an easy target.