I have looked out for my mom's health. I look out for my mom's health.

"I have looked out for my mom's health" wouldn't be incorrect, but there are implications to the choice of verb tense that leave questions in the native English speaker's mind. After reading it, I think "When? In the past but no longer? Until what point in time? Have you stopped? If so, why did you stop?"

Abe looks out for Jane, as she brings her to the playground to joining the other kids. Abe looks out for Jane, as he brings her to the playground to join the other kids.

Doesn't sound totally natural, as 'to look out for' has the nuance of 'protecting from a bad outcome' as opposed to just taking care of them by doing positive things like this.

Abe looks out for Jane, as she brings her to the playground to joining other kids. Abe looks out for Jane, as she brings her to the playground to join other kids.

if it is expressing purpose or intention ie she goes there because she wants to join other kids, "to" is better.

Abe looks out for Jane, as she brings her to the playground to joining the other kids. Abe looks out for Jane, as he brings her to the playground to join the other kids.

"Abe" is typically a male name. "The playground joining other kids" sounds like the playground connects human children together.