Is the opponents card order the same for me as for him

Are there some common rules on which cards to discard immediately?

That's pretty hard to say because it really depends on your build and playstyle. But the cards that cost 5-10 mana should in most cases be removed without thinking. Unless they're some really powerful high quality cards in which case it could be smart to keep one but only if your other cards seem good.

I think it's not worth risking early game by keeping high cost cards unless you're sure you can pull it off.

Besides the mana cost, which factors should I take into account making this decision?

Spell cards:

Some spell cards are more effective late game, even if they cost low amount of mana. Rather remove them unless you feel like they might come in handy early game. Keep good early game spells, especially those that cost low amount of mana and are able to destroy enemy monsters.

Minion cards:

I'd suggest keeping good taunt cards that are up to 5 mana cost. Taunts are very useful and you never know will you get one when you'll really need him later.

Low cost minions that let you draw more cards in any way should be kept as well. Quite useful.

In the end there's no magical formula which will tell you what to keep since you can't know what you're going to get next by drawing. Sometimes you remove cards you don't need early game just to get them replaced with other un-needed cards.

First of all, it depends on the deck type and your hero choice. Warlock's hero power has built in card draw that eliminates much of the need for card draw mechanics on cards. If you are playing a slower late game deck, you might be able to get away with slightly less card draw as your hand will be filling up naturally.

Card advantage is extremely important to win in Hearthstone. If you run out of cards, you are limited to playing only what you randomly draw that turn, rather than picking the best card from your hand. This severely limits your options to deal with your opponent.

Watch how many cards you play each turn, every turn. Each time you play more than one card in a turn, and only replace them with the single draw, you are losing card advantage. The more mana you have, the more cards you will be able to play each turn. Note which turn you run out of cards (or have nothing valid to play). Also note how many cards your opponents have on these same turns.

Then increase the amount of card draw in your deck until you can keep up with the pace of the game.

Finally, cards that perform an action/summon a creature and draw a single card (also referred to as card cycling) are not the only answer - more efficient card draw mechanics can come from cards such as Northshire Cleric, Gadgetzan Auctioneer, Acolyte of Pain, where more than one card can be drawn from a single card, increasing the value.