You can know the cards in the deck, but not what you'll draw
New information on the Deck has been published since you asked this question. The Book of Many Things gives a more detailed explanation of how the deck works under The Deck in Play / Handling the cards:
The Deck of Many Things has the power to transform anyone who draws from it, but to trigger its effects, the individual handling the cards must state their desire to draw from it, as well as the number of cards they intend to draw. Otherwise, the deck's magical powers lie dormant. Anyone who identifies a deck learns this about the deck in the process, per the rules for identifying magic items.
This means that under normal circumstances, characters can handle a Deck of Many Things safely. A character who finds a deck can look at it, spread the cards out, and otherwise manipulate the cards without triggering their effects.
This would indicate that you can look at all the cards, even without an identify spell, and therefore know what cards make up the deck. So this is clearly a knowable property that identify should be able to discover. However, what you will not learn is which card you will draw when you draw, because the deck states (DMG, p. 162):
Before you draw a card, you must declare how many
cards you intend to draw and then draw them randomly
That is, even if you as a character looked at all the cards before, and ordered them to put all the good stuff on top, once you draw, the card you will get is random. It's magic.
Partial decks
The solution to the problem of how that works if the players remove the negative cards first is simple: only when the whole deck is assembled, can its normal magical function be exercised. For example, the book suggest you can let players find individual cards separately first, and have these cards have some minor magical effect by themselves, but not the effects described in the deck. Then:
Eventually the heroes assemble the entire deck, which they can then use in the traditional manner, drawing transformative cards from it in a fitting climax to the campaign.
Stacking the deck
The Book of Many things also explains many other ways to use the deck, including that it is up to the DM to pre-determine the order in which you will draw cards "randomly", as long as it fits to their vision for the campaign, and says, addressing the DM:
You can ensure certain outcomes by stacking the deck with cards placed in a specific order.
So, you should be able to learn the number and names of the cards in the deck. When it comes to how much detail about the deck and effects of each card you further can learn, as this answer states, the rules do not specify further how to interpret what is included in "the item's properties" that you learn from identify, so that will be up to the DM.