Alright, first things first.

I'm a bit new to this game, summoner level only around 20-ish. That being said, I think I have a firm grasp on builds/strategy, can last hit very well, zone with the proper champs, never push my lanes, ward, etc.

However.

I cannot deal with Morde if I'm in a lane against him. It's like this giant hunk of metal got rammed into my skull which reduces me into a gibbering pile of soon-to-be-ghosty goo at his stupid looking spiky feet.

I would looooooooove some suggestions.

Harass him a lot if you're ranged, the big grey bar drops like a rock.
Buy a QSS if you're the carry, and ignore him in teamfights. Kill his allies first, since he can't do jack **** about it.

Let him push the lane, To harrass you and keep his shield on he has to do it. So get a gank, if yo ucan't don't stay anywhere near your minion, that way he has to choose between shield + farming or damaging you.

Poke champs win the lane against him.

Super EASY counter to him : Caitlyn. Play her and you'll see how he will drop very quickly.

hes annoying, get a gank on him and try to stand outside of your minions, problem with being lower summoner level is that not that many junglers in your games and so ganks are somewhat uncommon

or you could try to harass him when his shield is down

from personal experience, make sure you dont stand in the creep wave, try to clear the minion wave as fast as he is (so you dont draw minion aggro) and harass him as much as you can once he uses most (all?) of his abilities to build up his shield. he wont be able to use any skills for a short period of time and usually you should be able to break his shield and hurt him a little

If you're playing a carry, get a Quicksilver Sash pretty early on and learn to activate it as soon as you hear the laugh of his ulti. If you're playing as anyone else, get some MR anyways. Try to avoid getting into a 1 on 1 fight with him ever, and never ever focus him in a teamfight. Once late-game hits he's just a big ol' damage sponge unless he's able to ult the carry. If that happens, you run away.

What do you play with?