Is it not possible to searching for only even nonces? Of course you risk the chance of missing the nonce, but you gain an effective 2x hash rate.
Yes. It is of course possible. You are allowed to modify the whole mining software if the resulting block is a valid bitcoin-block. But you don't gain a 2x hash rate. It doesn't make any difference wheather you try all numbers between 0 and 29 or you try all even numbers between 0 and 59 because they are still only 30 numbers.
What if you search downwards for a nonce?
As I said: You could do that. But it doesn't make any difference in average. The normal case: The difficulty is very high. So you need to try all the nonce-numbers (0-4.294.967.296). If no one else mined the block, you can change the timestamp or modifiy the coinbase-text and try all the nonces again and again because the hashes will be completly different then.
Skipping (even, odd, every third, etc).
There is no advantage. It's actually a little disadvantage because you need to calculate
nonce++ 2 times instead of 1 time or you need to do the if statement
if(nonce%30==0) or whatever instead of doing nothing. So you would waste energie, time and mining power instead of hashing a block.