Edit: As promised, here is a quote from the Yu-Gi-Oh! tournament Policy regarding altered cards, emphasis mine, where we can see that in fact marked cards may not be used in tournaments:
Cards that have been altered from their original appearance with surface decoration may be used for play only if the alterations do not obscure any portion of the card text, make the illustration difficult to recognize, or make the card distinguishable from other cards in the Deck while it is face-down.
The Head Judge of the event has final arbitration on whether or not altered cards may be used at the tournament. Duelists are responsible for determining whether or not their altered cards are acceptable for use before the start of the tournament.
Cards with any markings added on the front or back, may not be used in any Yu-Gi-Oh! TRADING CARD GAME World Championship tournament
In casual and friendly games not, however on official games or tournaments:
Are there any rules against this? Does it make the card in any way illegal or unplayable?
In a way, it could depend on what that mark is (that is, if it is noticeable by others). But anyways, all marks on cards are not allowed on official games.
In fact, when you assist an official tournament Judges first go through your Deck list, checking you indeed have and use those cards.
While on that, they also check to see if any card you own has marks or is in bad shape. They do this as marking cards is considered cheating, as you could easily stack or track them, or do some other dishonest tactic.
If they find any such card, damaged or marked, they will ask you to get a mint, near-mint card to replace it, or well drop it from your Deck.
If, during the tournament you are caught with marked cards, it could mean game loss for you, or even dropping from the contest.
As a bonus: can I do this to the (inside) of the sleeve instead?
Again, could give room to cheating and stacking cards. Having a thicker card will make it stand out from the others, and you could strategically place it on top of your Deck or similar deceptive moves.
Furthermore, card sleeves must be identical on all your Main Deck cards, and also be in good shape. Cards on your Extra Deck should be in the same condition and have a different color that your Main.
That being said, marking your cards will also affect it's value and price, so you may be losing money if doing so.
I'd say it's on your best interest to learn your archetypes and card rulings, instead of having to mark your cards. Eventually you will come to know them perfectly, and will regret having marked your cards.