I’m a former hotel employee as well as inspector for an international chain. Here’s what I can share with you.
We use(d) the terms “stain” to mean something that can be cleaned. “Discoloration” meant that it was permanent. I’ll use these terms in this post.
In the second photo, the grout appears to be stained. Most likely, it can be cleaned with with a chemical product to bring it back to a “like new” condition. Finding the right one can be challenging. Some are too strong, some are too weak, and some are just right. When it becomes this stained, it is going to take elbow grease and a scrub brush (toothbrush size works well) in order to remove it. A pressure washer is another tool that can work.
The problem with strong chemicals and power washers is that they can damage the grout and/or tile. The best way to combat it is through wiping down the tiles after use. It also depends upon the grout. Today, the types vary greatly. Most now come with a sealer mixed in. In the past, they didn’t and were more susceptible to build-up.
In the third photo, if the clean looking grout was cleaned while the other half wasn’t, then it is definitely a stain and not a discoloration.
In the fourth photo, yes, it’s mold. The grout can most likely be cleaned. The caulk is another matter. Some of the mold may come off with cleaning, but it discolors easily. It may be time to strip and re-caulk. And caulking is an art. You can most likely learn the right methods through DIY posts as well as practice.
As for a more permanent fix for the grout, there are a couple. Based upon the tile size, they are probably old (but not ancient). The grout most likely doesn’t have a sealant. Once cleaned to your satisfaction and bone dry, use a grout sealer finish on the grout. It’s a painstaking process, but can be worth the effort in the long run.
The other option is to have the tub/shower stall wall tile replaced. It will still require regular cleaning, but not nearly to the extent that older versions do. Just make sure that the tile grout has a sealer included in its mixture.