It seems that all your problems arise from standing still.
I think you are saying that you find it difficult to last hit because when you stop moving your champion will automatically attack. You can stop this by pushing "s" which will stop your champion from moving and stop them from automatically attacking until another command is issued. So you can push s and then when the time is right attack for the last hit on a minion.
A better alternative however is to constantly stay in motion, move back and forth, side to side and attack the minions when you need to. This has the advantage of making it harder for your opponent to hit you with autos or skills and also makes it possible for you to manoeuvre into positions from where you can safely attack and harass your opponent.
Aside from that, it just takes practice. Of course you can also make it slightly easier by having flat damage runes but it always comes down to practice as you'll have to last hit on low ad champions like lux as well.
One thing that may help is to note your AD and then note what the minions health bar looks like when their hp is below your ad. As you're not going to be clicking on every minion to see their health this can be useful for helping you get used to what a killable minion's health bar looks like. Remember that the ranged minions, melee minions and siege minions all have different amounts of hp all of which scales throughout the game.