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The entry:
returns the limit 0. And the message about that "surd" was temporarily changed to a fractional exponent. But the entry:
returns -6 and the same message about "surd". Am I doing something wrong with the limit command?

The warning is the same but it is not sent by the same command: it's by limit in the 1st case and by simplify in the second case. Both will replace surd by x^(1/3) in intermediate computations.
simplify will replace your expression temporarily by (x**2-1)/(1+x^(1/3)), then simplify then replace back x^(1/3) by surd. Then evaluating at x=-1 will return -6.
limit will replace surd by x^(1/3), then since (-1)^(1/3) is a complex != -1 the limit at x=-1 is evaled to be 0.

What surprises me is that this is the case even when I don't have complex calculations activated. What if I want to use Xcas in a real analysis course? Are there any settings to restrict calculations to real numbers?

I would recommend to use fractional power instead of surd, like this near -1
limit((x**2-1)/(1-(-x)**(1/3)),x=-1)

Thanks!