To avoid the airflow when the cymbals are closed, try placing a cardioid microphone slightly above the cymbals (10 - 15 cm) pointing at the middle. Since the signal primarily consists of high frequencies, a low frequency roll-off could be applied in the mixing console to keep the overall sound bright and crisp.
Omnis can also be used for this application, but they need to be placed a bit closer. The DPA 4007 is a good choice, but also the compacts DPA 4036 or 4037 can be used. The latter are very small; mounted in the compact shock mount makes positioning easy and gives a classic look. In certain music types such as Jazz, Blues, or Folk, the Snare and Hi-hat can be picked up by one strategically placed omni mic. The placement will be between the Hat and Snare and you will have to find a placement balance point based on each drummers technique. An omni such as the 4007 or 4006 does this nicely and yields very good "air", but with a little more effort the 4011 or 4021 cardioid can also do the job and produce an even tighter result.
The drumkit can produce very high peak sound levels. Levels in excess of 120 dB at a distance of one meter and at a few cm from a drum or cymbal head 140 dB or more is not unusual. It's obvious that the microphones must be able to handle these levels without clipping, which is not always the case in many recording situations.