OK ^^^ What he said^ Bobby, you didn't answer his question, you just talked about favourite compressors around here.
1. If it's a good one, it will compress your signal, leading to a smaller range of dynamics with your playing. In other words, it will bring down the peaks and bring up the valleys.
2. The drawbacks are that you have a smaller dynamic range to work with if that's important to you. Also, some compressors are said to 'colour' your tone, or in other words, people don't always like the way they sound.
3. Depends on what you mean by 'play nice'. It's an effect. All effect pedals interact with each other, so really it depends entirely on the circumstance.
I will note that you asked about what it will do for your rig, and if it'll play nice with pedals, but you didn't share anything about your rig. That would be helpful info for us to answer your questions.
Jacob
What will a good compressor pedal do for my guitar/amp rig?
What are the drawbacks to using a compressor?
Do they play nice with other pedals?