another problem with crossing the border into Panama to renew your visa is that Panama recently changed their visa requirements. They only offer a 30 day tourist visa with the option of renewing the tourist visa for a 90 day extension, one time only. So, if you are considering crossing the border to renew your tourist visa in CR, it would be better to use the Nicaraguan border. However, you can't bring a car across the border into Nicaragua.
You definitely will have problems with bringing your dog. Dogs in CA are mostly left to fend for themselves and there are dog gangs...as the poster above stated. It is very difficult to be a perpetual tourist in Central America. We tried it for a year and now all of the countries are changing their visa requirements making it difficult to border hop every 30-90 days.
Can you get a work visa for teaching English through the school?

Well, I'm not saying the above isn't true to a point. The high taxes are a problem (33% of the value of the car by their books). But the dog problem I think is at least overstated, at least in the less populated areas down south. Folks accept our current dog everywhere we go (he follows us up the road so we just started bringing him with us anyway). Some restaurants eventually shoo him outside.
The border deal is that you can supposedly only bring the car out and in twice in a row (every 90 days). And, you can get the car impounded for being over the limit. It is however, quite possible that you could hire someone (a recommended someone) to fix up your papers at the border for a fee. Don't just show up and try it on your own, I would think, but maybe it could be done. If not, there is some rule here about wheather you started the paper process yet or not. As long as you started the paperwork, it might be considered "limbo" for the interviening months that such things require.