since this forum is about plumbing, I'll add what I can about the big mess in the first picture.

There are two places in the center of the picture where you can see the "person building it" used short-radius fittings. These are 1/4 bends, i.e. 90 degree turns. (They come in short and long sweeps.) The use of short radius bends in drain plumbing is a big warning sign, and it often means that everything else is terrible too.

A second big indicator that "everything" is wrong, is the use of Wye's and 45 degree slopes -- far overdone. Using San Tees is better for drain plumbing; there are situations where a Wye is the right fitting, but not here, not to create these 45 degree slopes down to the next 45 degree slope.

It usually corresponds to a lack of venting. And, then, looking over to the left we can see a P trap with NO vent and worse yet, a slope far exceeding the 1/4" per foot recommended. (Slopes can be more, or less, but that has to be included in the arithmetic whenvents are planned).

I'll stop here. More later, if we wish to discuss and plan a plumbing system.

about the joist: it is true that the knowledge required to find out what to do to reinforce a single joist is available on the internet. A homeowner can calculate it too. Assuming someone gives him/her the right key words to search on.

David