I don't do GS, and have no idea about the definitive answers to any of your questions, but since replies do not seem to be particularly forthcoming I'll post some vaguely relevant sociological ramblings.
1) Not sure that there is (/can be) definitive agreement either way. For Marxist's, class will continue be a relevant concept as long as capitalism lasts (ie. until a classless communist society is established). However, for perhaps...a postmodernist, it becomes entirely irrelevant (along with most other forms of "categorisation") as people need to be taken on a more individualist basic and not assigned to rigid categories, etc.
I suppose it will also link with your answer to question 2: if there is no social mobility, then class is hugely important - the class into which you're born determines all your 'life-chances' (Weber??), etc. However, if there is a huge potential for social mobility then class becomes less relevant as a concept, since people's class can change and therefore it cannot be used so accurately / usefully in social and political analyses.
2) How on earth is one supposed to answer this? "Yes, we have...er...26% mobility over here...how's America doing?". Ridiculous thing to try to measure (though I'm sure someone has). Maybe you could look at the opportunities for social mobility: how can one change one's class (presumably for the better)? Education leading to good jobs money and status (though it may take a couple of generations), or other options...marrying into upper-class families, etc. Do things like the National Lottery increase social mobility? People winning that amount of money all at once have the potential to drastically alter their lifestyle (but is class more than lifestyle? - attitudes, "new money", etc.).
3) Damn. See above.
4) How are you defining class? Education, income, job status, attitudes, lifestyle, etc? Some of the above, if not counted under "class", could be alternatives. If we're talking on an individual level, everyone uses different differential criteria; if we're talking on a meta-narrative theorising level, look at alternatives to Marxism (feminism: sex not class, etc.).
Not sure I really understood the questions but nvm!
ZarathustraX