On this page you can find frequently asked questions: "Miking up and recording a drum kit".
In terms of sound capability, a large diaphragm condenser is perfectly able to record the low, large soundwaves of the kick drum, but as condensers are much more sensitive than dynamics, its important that it has some form of attenuation to avoid overloading. Most good quality condensers will have something like a -10dB/-20dB pad switch that can be used to prevent the kicks transients from causing distortion. With cheaper condensers, manufacturers sometimes save on cost by leaving out this feature as theyre often used exclusively for vocals which generally dont produce nearly as much level!
There are two main ways to attack this problem, and you may wish to employ both methods at once. The first is to place a blanket or other acoustic insulation over the kick drum to keep the spill out, much in the same way as you might do with a piano - this will dampen the high frequencies from the cymbals in particular. The second method is to insert a noise gate into the kick drum channel, adjusting its controls to eliminate as much spill as possible, without the process becoming too obvious, or losing the quieter notes altogether.
Cheap cymbals can be harsh and tinny so you may need to consider upgrading them to some smoother sounding instruments. A good set of cymbals can make a great investment for a studio. Failing that, make sure that the mics your overhead mics are not too toppy, and try replacing them with a pair that you know has a flatter response. Also, try listening to the overheads with the tom and snare channels switched in, as the overall mix may reveal a more acceptable sound.
In short, yes - their figure-of-eight pattern makes them especially suitable as overheads when you have a good sounding room, and they can usually handle high SPLs without distortion. The warmth of the ribbon means that cymbals will generally sound less harsh in comparison to condensers, and the natural, open sound often allows you to get away without miking the toms.