Your chances of getting a double down are reduced if there are more tens in the deck. That's just common sense. Obviously, if you do get the double down combination in a high count situation, then your chances of getting the ten on top of it is greatly increased.
As far as splits go. Other than splitting tens, your chances of getting split numbers will not be great either. And most alterations in play based on a very high count, only call for the split of tens against a five or six, which isn't gonna happen all that often either with a remaining shoe full of tens.
Are you more likely to win more double downs and split tens with high counts? Absolutely, but you are not gonna get more of them compared to low count situations. Even if you factor in the alterations in play (splitting tens, doubling on 11 vs. A, doubling on ten vs. A or ten, doubling on 9 vs. 2 and 7, doubling on 8 vs. 5 or 6)