Significado de "My mother had me tested"

Hi there!
I will try to help you:

"What someone does for us"

We use have + object + -ed form when we talk about someone doing something for us which we ask or instruct them to do. It emphasises the process/action rather than who performs.

In English we use HAD (have or has) to emphasise that something has been done by someone else.
Examples:

I had my hair cut. (I didn't do myself)
She had her car washed. (she paid for that)

In some contexts (informal) GET/GETS/GOT can be used instead of HAVE.

In the show you wrote about Sheldon says: "Minha mãe me testou." or "Minha mãe fez com que eu fosse testado." 'course she didn't do that herself.

Cheers!