I seem to have trouble playing Zed mid lane without Death Mark. I've heard that Zed should use his 3 abilities as much as possible at level 3 to maximize his strong early game. If this is true, then how should I proceed with such a combo? In which sequence should I use my abilities?
You could try to place your shadow in front of you, slow him with your spin and then throw your shuriken together with your clone counterpart. Try to hit both of them on the latter one for safe damage trading.
Also, if possible, you can try and land some skills on recharge between the lasthitting instead of waiting for the full cooldown.
EDIT : Ok before ult but considering you will be lvl 6 at around 6:00 => 6:30 (without dying) you will have only 4:30 of effective laning without ult .... anyway just W (near target)=>E =>Q to try to harass as much as possible without taking damages and waiting the lvl 6.
Last hit avoiding as much as possible to take damage(use your Q OR W/E Combo) dont push your lane and w8 your jungler ;-)
After getting ult, whenever you have ult you should have a kill ... at least on squishes
On mid lane try to harrass as much as you can and look for an opportunities ;-)
But what is an opportunity ? Its when ever your lane opponent has used a skill that he could use to prevent you from comboing him like Lux/morgana snare Kah'zix jump, Elise/sion stun, etc ...
The point is, when you use ult you get untargetable so if you can dodge the right skill ulting your oppenent he will have that on cooldown.
Ult Combo :
launch ult => E to double shadow slash(big slow) => Q with your shadow for increased damages => as much AA as you can ==> W Living chadow BEHIND the target ==> E Shadow slash (slow again) ==> W Swap ==> AA as much as you can ==> BOOM Ult Mark trigger ==> KILL :-D
You can add Ignite if you need additionnals damages.
An other good use of the ult is to counter dive. don't forget you are untargetable so you can doge some CC and you can swap using your R and W to brain the diver running around the tower
I think that you could try laning them by shadowing them whenever they get close and if they pass it switch to avoid skillshots or mid cc (i.e. Annie's stun)