No, it's not. There is nothing intrinsically expensive about chemical propulsion.
The expense is a function of the actual cost of the chemical propellant & infrastructure enabling it’s use.
The cost of chemical propellants is trivial, on the order of one percent of the total launch costs. All one needs to do to get costs down is to increase flight rate and stop throwing hardware away. SpaceX is working on both.
SpaceX vehicles are a first step, but keep in mind they are heavily subsidized through NASA.
No, they're not. This is a myth.
Without U.S. Government intervention, how long do you suppose they would remain in business?
A long time. Indefinitely. There is no "government intervention," other than purchasing services from them. And they have a large commercial backlog. They just signed a contract with Intelsat.
Funding
As of May 2012, SpaceX has operated on total funding of approximately one billion dollars in its first ten years of operation. Of this, private equity has provided about $200M, with Musk investing approximately $100M and other investors having put in about $100M. The remainder has come from progress payments on long-term launch contracts and development contracts. NASA has put in about $400-500M of this amount, with most of that as progress payments on launch contracts. SpaceX currently has contracts for 40 launch missions, and each of those contracts provide down payments at contract signing, plus many are paying progress payments as launch vehicle components are built in advance of mission launch, driven in part by US accounting rules for recognizing long-term revenue.