Not quite.

One of you takes the child through (in your arms). If no alarm, both are clear. If there is an alarm, both will be screened (adult with handwand, child most likely with a quick pat-down).

If one of you is SSSS and the other isn't, or if one of you will *definitely* alarm (artificial joints, etc), have the "clean" parent carry the child through.

Jackets will likely need to be removed, but shoes shouldn't (unless there is metal, or possibly light-up shoes with electronics).

Stroller will have all bags removed and sent through the X-Ray, then be folded up to go through. Wheels up will usually help. If it is too large to fit, a screener will physically inspect it.

As an aside, some airports may have a special line for parents with small children or elderly passengers needing assistance. You might want to look for that (though not all airports have them). I know at the airport I work at, *most* screeners will try and offer some assistance to parents with small children, especially mothers travelling alone.

HTH!