The bulbs on mine are also mounted too deeply within the red lens to get out, but they are mounted to a plastic slider that pushes up into the housing and clips in. If you pull the slider away from the housing at the bottom the whole apparatus slides out.
If I needed to replace my lights/housings, I would seriously consider using LED taillights. The LED’s are much more reliable than conventional bulbs as the light is not generated by a resistive heat element. I’m thinking I’ve seen them around at stores for around 50 bucks or so, and I would expect that they would be the last lights you’d ever have to buy (lifespan of an LED is, for all practical purposes, infinite).
As far as the fast blinking, it could be one of two things. Whoever posted that it could be an out light was correct in that some vehicles will flash faster when a light is out. If it only does this when the trailer is connected, then it is the flasher relay. If the ‘off’ time is constant, but the ‘on’ time gets shorter that’s a dead giveaway.
The stock relays most cars come equipped with use a timer circuit that relies on the amount of current that your bulbs take. This current decays a charge and kicks the relay off. When you add the trailer lights to the chain, it takes more current and dissipates the charge faster (thus disengaging the relay at a higher rate).
Any auto parts store, and even K-mart or Wal-mart should have a ‘heavy duty’ flasher. These aren’t really any more heavy duty per-se, but they rely on an internal timing mechanism that’s not dependent on the amount of current they pass – thus a constant blinker speed regardless of the electrical load.
On most cars today, the relay is in the fuze box and looks like a tin-can like Herb described. On older cars, you’ll have to look up under the steering column and dig around in the wiring harness to find it.