They most definitely do. A good price is a good price... usually when there's a system failure and quotes freeze, a competitor market maker does update their quotes and eventually hits the others. But it will depend a bit on spread etc. Also, if there are stale/frozen quotes... they will usually appear when the underlying moves in one direction... the risk for other market makers is then, if they hit all of the frozen quotes, they will have a significant delta position and likely can't get out of that so easily. Sometimes this happens and markets makers let it go, they might get called by the exchange and forfeit the trades. All market makers are competitors, but also colleagues... so in the end there's no point really to hurt others. At one point in time, you yourself might need a favor... and it's better to have some decency than to f%$k someone else over.
Yep. Usually you quote around your theoretical value, which is derived from your Implied Vol which is your main parameter to adjust pricing. But, if you have a large number of options in a specific strike, you can quote slightly towards the opposite side to maybe get lifted more quickly... downside is that that way you also price it lower, and therefore eventually on paper have a lower value... so bookvalue will start to be lower as well, or so it should.