Hey there,
I have found some answers for you in the fact that I've experienced some of these issues myself.
As far as the parking brake latching, or, not latching as the case may be, take your Breeze to a Plymouth or Chrysler dealer in your area. This issue is actually part of a 2 part recall. One is the floor shifter not locking into park with the ignition off, and 2 of course, is the parking breake not latching properly. Both of these issues are actually due to a design flaw in the center console.
As far as the check engine light goes, check your oil. These vehicles were actually designed with a low oil warning other than the normal light on the dash. Whenever the oil is low, the check engine light may light up first as a warning before you're running on extremely low or no oil. Also, beadvised, as with all 4 cylinder engines, the 2.0 will burn through alot of oil when driving through hills. If you live in a hilly area, it is recommended that you use 10W 40 motor oil instead of 5W 30. It may not be as "energy efficient," but, it'll save your car's engine.
As far as the dash goes, and as far as most digital dashes of the late 80's and early 90's, or, at least, their digital parts, these dashes were no different in the solder joint failure area. This actually should have been a recall considering the number of vehicles it affects including the Plymouth Breeze, Dodge Stratus, Chrysler Cirrus and Siebring, but, unfortunately was not for a couple of reasons:
First of all, most vehicle BCMs were designed by this time to store the vehicle's milage instead of the dash cluster storing it. GM even took this one step further by making the cluster error out if it appeared that the milage was in an attempt to be tampered with. Moving on... If you notice, everytime your cluster comes back on from going out, the milage has changed, so Chrysler saw this as a "no harm, no foul" type of situation.
Secondly, Chrysler did not make a recall on this because no one was ever killed in a vehicle due to the odometer and/or tachometer not working. It was considered just an annoyance, and not a safety concern unlike the parking brake and floor shifter.
With that said, if you take the dash to a Chrysler, they will charge you for a whole new cluster, or at least the repair.
Best advice for this is to do 1 of 2 things: 1: If you have about 2 hours of free time, go to your local Radio Shack and pick up a pen soldering iron meant for electronics and a tube of the thin electrotek rosin core solder. Also, if you don't have it, get a multi head torque screw driver, you're going to need it and follow these instructions.
DO NOT get or use a high powered soldering iron for this can damage the board and/or components.
Remove the dash top and remove the cluster using a simple phillips head screw driver, and take the cluster apart using the torque driver.
Remove the backboard, and the front plastic window and cowl, and unscrew and remove the just the speedometer, tachometer, and odometer combo panel.
On the backboard, go ahead and first re-solder every place where it looks to have a component making contact with the board via the solder. most of this will be done on the bulb side up side of the board, but, if you wish, you may also re-solder the loop-connections on the other side of the board as well. This is a very simple, but tedious task and can take an hour.
On the back of the combo panel, re-solder every contact point or solder joint behind the florescent display as well.
Then, simply re-assemble your panel and voila. Your instrument cluster is working once again as if it was brand new or better. This also gives you a chance to clean the cluster inside and out if it has become dirty.
This method will cost around $10.00 or less to do.
2: You can take the cluster to an electronics repair shop which may cost you up to $100 at least.
DO NOT replace the cluster with one from a junk yard or ebay for you may be right back in the same situation. This way, you know it's fixed atleast.
In the 2.0 engine, carbon build up is extremely common and can cause the vehicle to smoke and even fail an emissions test. Best way to repair, put either some all system cleaner or injector cleaner in the fuel tank every time you do a tune-up and the smoke problem will no longer exist.
It is very important to do this because, eventually, it may clog the catyllitic converter.
Well, I hope this advice helps you all out.
Thank you,
Starflare5.