"Home production" is the econ term for everything that might be considered "housework" or "do it yourself", not building houses. The problem with excluding it is that the same service done by a contractor or employee would count towards GDP.

The leisure problem, is that leisure time is a consumer good, but it's free and we have no good way of pricing its value to the consumer.

"Real GDP" has all of the same problems as GDP. It just has a very flawed inflation adjustment applied to it so that comparisons over time make more sense.

I'm not an expert on "Gross Output", but my understanding is that it captures more of the structure of production, by including intermediate stages. There are pro's and con's to doing this. The con that most people will raise is the supposed "double counting", since the earlier stage prices are supposedly included in the price of the final goods. The pro's are that you don't have to make arbitrary decisions about which goods and services are final goods and services vs which are intermediate. It is also more of a measure of overall economic activity. It does move with GDP most of the time, but it generally declines more sharply during recessions.