There has been very little or no research done on whether a salt cave can reduce stress levels (or many of the other supposed benefits you list). But, given that they generally make you recline in a darkened room and listen to relaxing music, I would say it's plausible that you would be more relaxed afterwards - but that says nothing about the efficacy of the salt itself. They don't describe the mechanism by which the salt works, so it's not really possible to study it. There are, however, people (including the site you linked to) that claim that salt aerosols and/or going into a salt cave can help with respiratory problems.
Using salt as an aerosol in a cave is known as either halotherapy (salt therapy) or speleotherapy (actually going into a (salt) cave). These are relatively new treatments in the west but have been practiced in Eastern Europe and Russia for several years.
A Cochrane review was conducted on Speleotherapy for the treatment of asthma, and of all of the (3) available trials meeting their criteria they found only one which did not have a flaw serious enough to warrant its exclusion.
Three trials including a total of 124 asthmatic children met the inclusion criteria, but only one trial had reasonable methodological quality. Two trials reported that speleotherapy had a beneficial short-term effect on lung function. Other outcomes could not be assessed in a reliable manner. A further search was conducted in July 2000. One further paper was excluded (see excluded studies)
The available evidence does not permit a reliable conclusion as to whether speleo-therapeutic interventions are effective for the treatment of chronic asthma. Randomized controlled trials with long-term follow-up are necessary.
Asthma Australia has warned against salt therapy, stating that it may actually trigger asthma attacks:
A respiratory physician and chair of the Asthma Australia medical advisory committee, Simon Bowler, said he would advise asthma sufferers against undergoing salt treatment.
''If somebody inhaled significant concentrations of a salt solution it could actually bring on an asthma attack,'' Dr Bowler said.
Salt inhalation may have modest short term positive effects on respiration and mucus clearance rates for those with cystic fibrosis, but I would expect the amount of inhaled salt in this study (5ml of 7% NaCl solution, 4 times daily) is much, much greater than one would receive from lounging around in a salt cave for an hour.
I have searched for other studies on halotherapy and speleotherapy, but many of them have obvious conflict of interest issues that are not addressed, for example one study was authored by the employee of a salt aerosol manufacturing company. Similarly, I don't know how reputable the Russian "Journal of Aerosol Medicine" is. There be dragons here.