Welcome!
Yes, she needs to be earning £18,600 minimum a year to sponsor you (and needs to be in this job for a minimum of six months before she can sponsor you). I'm a bit confused about the kids - do you have kids you are trying to move over as well? If so, that increases the requirement. Any children she has will not increase the requirement.
To move to the UK without her sponsorship, you would need to be sponsored via a work visa. These are really tough to get. Are you in a highly specialised field? If so, you may be okay. Or do you work for a US company with offices in the UK (potential transfer)?
Unfortunately the £18,600 threshold was set to try to reduce benefits people can claim. Her benefits will be affected by earning more. That's not a bad thing though! She shouldn't be worse off.
Yes, if she doesn't make £18,600 you can use savings to offset the amount. Say she makes £15,000 a year. You would need:
£18,600 - £15,000 = £3,600 x 2.5 = £9,000 + £16,000 = £21,000 in savings for a minimum of six months to make up the shortfall. The savings can come from either of you.
I hope that clarifies a few things for you. Ask away!