The first thing that you need to learn is the difference between a two stroke and a four stroke.
The four stroke is type of engine you would be most familiar with, it is what your car engine would be. A four stroke engine has a "cam" and valves and a lot of moving parts.
A two stroke engine requires that there be oil mixed in with the gas. The advantages of the two stroke are that they are lighter, fewer moving parts (less maintenance / easier rebuilds) and they can produce almost twice the power for a given displacement as the four stroke. The disadvantages are that they smoke, they are not environmentally friendly, and the power they develop is very "peaky" which can be difficult to control.
You are not going to find a bike that meets all your "wish list" items. Let's talk about street legal for a second: what you are talking about is generally referred to as "dual sport". Many states do not allow two strokes to be street legal at all. In California it not only needs to be a four stroke but it needs the proper emission control stuff as well.
IF you ride this bike off road, you ARE going to fall. It isn't even the question of "if" or "when" it is "how many times" and "how hard". Mirrors and turn signals take a real beating in such falls. If you get a dual sport bike, do everyone a favor and remove the mirrors and turn signals before you go off road so you don't litter the trail with them.
Tires on a dual sport bike are another issue. To be safe and legal on the road the tires need to be "DOT" approved. Such a rating really limits your choices on tires, and basically a tire that is best for dirt won't survive long on pavement and a good pavement tire sucks in the dirt.
All the people I know who have dual sport bikes do NOT use them for commuting to work. What they do use them for is a brief transition from one trail to another using a public road. There are a lot of organized "dual sport rides" that requires a street legal bike, but on a 500 mile ride only 20 miles might be on a public road. Everyone trailers their bike to the starting point and back home.
Budget is another issue you will have, especially for a dual sport bike. To get a dual sport bike under $1500 you will need to buy something 20 years old, but if you do that you had better be doing all your own maintenance work or the repair shop bills will kill you.
The MX track is certainly not where you would want to take a dual sport bike. It would be too heavy and probably not have a suspension suitable for the jumps.
For non street legal, bikes will come as "off road" (also called "trail") or "MX" versions.
The MX bikes are full race, no frills. They will be lightweight, long travel suspension, high performance engines (either two stroke or four stroke), seat height of 38 to 39 inches and will be expensive. They will not have a headlight or even a kickstand. Examples of MX bikes are the Yamaha YZ line or the Honda CRFxxxR line (with Hondas you have to look at the last letters to establish what type of bike it is). A new MX bike will retail for around $6k to $7K. The high performance engines need frequent rebuilds.
The trail bikes are heavier, limited suspension travel, a four stroke low performance engine (you can get away with ignoring them for a while) and generally a lower seat height (34 inches). They often come with electric start and a kickstand. A new trail bike will retail for around $4k
Obviously, a used trail bike will cost a lot less than a used MX bike of similar years.
If you really want a dual sport bike then I would recommend a Yamaha XT250 or a Honda CRF230L, although you will have to be pretty old to get into your price range.
A good trail bike for you would be a Yamaha TT-R230 or a Honda CRF230F. You should be able to pick up one of these in your price range that is around 5 years old. These would be great bikes to start on but I suspect that you would want to upgrade a year or two down the road.
Don't forget to allow budget to buy all the support equipment and riding gear.
Must have:
Helmet
Boots
Googles
Gloves
Highly recommended:
Chest protector
Knee pads
Riding pants
Rod