Does 911 dispatch you directly?
most of the time, yes... sort of. almost all 911 systems have some type of emergency ambulance system. Sometimes the 911 center dispatches the ambulances directly; other times they will pass the call information (and often caller) to the ambulance dispatch center with an automatic computer transfer and/or a one touch phone transfer, who handles all communication between the 911 center and the ambulance. Smaller towns have 911 dispatching directly, larger cities have dedicated police, fire, and EMS dispatchers.
In Israel there is only 1 national dispatch that answers 101 (our version of 911 calls) and dispatches ambulances. Its a national ambulance service so no different companies.
That's not how it is in the US. Each town or county has their own Ambulance service, or contacts out with a private service or 3rd party non-profit company to provide EMS services to the area. Some places have multiple EMS agencies providing EMS services to a give town, city or county. In theory, all standards are similar.
Also, all of our dispatches are also field medics. This helps in getting the right info from callers or walking callers through cpr. Do the people who answr 911 calls also have ems training?
Sometimes... but not always. Some dispatchers are former field providers, and some are EMTs. However, there is not requirement to have EMS training, aside from Emergency Medical Dispatcher training (and even that isn't required in every state). Some EMS dispatcher centers require dispatchers to also be EMTs, but not every 911 call taker has EMS training
Can the dispatch the closest ambulance across state lines (like if the caller is in Nevada and the closest ambulance is in California)? So curious how it works there.
As a general rule, ambulances don't cross state lines, even if they are closer. In some areas, they won't cross county or municipal lines either, unless requested. There are historical reasons, funding reasons and political reasons for this, as well as regulatory reasons.
You also need to remember that Israel is the size of NJ, so it's not exactly a fair comparison. For example, MDA handles 2.7 million requests for service a year, and that's for the entire country; in NYC,
the FDNY EMS system handled 1.7 million request for service in 2018, and that's just for one (albeit very large) city. I know of cities that handle 75,000 calls for services just within their city limits. I'm not saying NYC's EMS system is anything to admire (in fact, I've ranted about it on this website), but there are certain quirks about the US system that make it an apples to pineapples comparison to how other countries do things (not saying its right or wrong, just different and hard to replicate)