If my daughter injured her friend at our house, would homeowners insurance cover this?

Yes. Generally speaking, homeowners' insurance provides coverage for liability and medical expenses arising from bodily injury sustained by a third party while on the insureds' property. However, it is possible that your insurance company may attempt to allege that this act is excluded from coverage because it was an "intentional" act. Intentional acts are excluded from Texas homeowners' insurance policies. Regarding your second question of parent-child liability, as a general rule minors are civilly responsible for their own torts. However, a parent can be liable for the acts of a child under various theories of liability including but not limited to negligent conduct, willful & malicious conduct, and a duty to control the child.