Is it okay to draw from the middle of the deck?
Mathematically the probabilities are unaffected. Let's consider a simplified example with just 4 cards.

Let's say we draw one card from the 4 aces from a standard deck of playing cards. The probability of drawing the Ace of Spades is 1/4.

Now let's split the deck evenly and pick the first card from the first split to compare the probability of drawing the Ace of Spades. Using the Law of Total Probability, there is half a chance the Ace of Spades is in the first split of 2 cards. There is also half a chance that in the first split, the first card is the Ace of Spades.

Multiplying these out gives the probability of drawing the Ace of Spades as being (1/2)*(1/2) = 1/4, just as in the case of drawing normally.