FluentValidation: Check if one of two fields are empty

You can use When/Unless condition:

RuleFor(m => m.FirstName).NotEmpty().When(m => string.IsNullOrEmpty(m.LastName));
RuleFor(m => m.LastName).NotEmpty().When(m => string.IsNullOrEmpty(m.FirstName));

or

RuleFor(m => m.FirstName).NotEmpty().Unless(m => !string.IsNullOrEmpty(m.LastName));
RuleFor(m => m.LastName).NotEmpty().Unless(m => !string.IsNullOrEmpty(m.FirstName));

As for your second question, FluentValidation works with client-side validation, but not all rules are supported. Here you can find validators, that are supported on the client-side:

1. NotNull/NotEmpty
2. Matches (regex)
3. InclusiveBetween (range)
4. CreditCard
5. Email
6. EqualTo (cross-property equality comparison)
7. Length

For rules that are not in the list you have to write your own FluentValidationPropertyValidator and implement GetClientValidationRules. You can find a few samples of this on the StackOverflow by doing simple search.