Age often makes a difference here. However, in the early years, I find that geldings are easier to handle because they aren't as distracted (if you know what I mean, lol) which makes it easier to keep their attention to work with them. Stallions and mares alike are usually concerned about who's in charge. Geldings only care if they're fed first, lol. With that it's important to know that every horse's personality is different, so it's not a one size fits all type thing. A gelding is viewed as the safer horse, but the fact is we won't know what a horse will be like after he's gelded either. Also, it's important to know that the age a colt is gelded makes a difference on whether he acts like a gelded or a stallion. Best to geld as early as possible to calm the boy down, but only if necessary.
Only when a racehorse finishes his career will an owner seek to put it out to stud. The fees vary. more or less. However, for the very best winning history stallions it can be more than $100,000 each time. A select few can earn much more. For example, a race horse name Storm Cat stud career at his peak was $500,000 each time.