It's always fine to look at unstandardized coefficients. I don't typically use standardized coefficients for much. Comparing the values of different coefficients to each other and interpreting that as a measure of 'importance' (which is what I think you're getting at) is a dicey affair. To do so, you need to assume that the units of the different variables are commensurate (if X1 is meters and X2 is seconds, how many meters equals 1 second?), which is usually not reasonable. The fact that your variables are all scaled 0-10 makes this look less suspicious, but doesn't necessarily make it better in actuality.
A different issue is whether your outcome variable is also 0-10. It sounds like it is. If so, standard (i.e., OLS) multiple regression may not be valid. If the Y data don't approach the bounds and the residuals look normal enough, you should be fine, but otherwise you may want to do ordinal logistic regression.