Like others have said, I'm an advocate of leaving the tags out when it makes sense. Bardrothen's first example set above demonstrates how you can establish who's speaking and give a better description of the whole scene at the same time. It's a terrific way to help the reader visualize the scene and avoid the repetition of "said, said, said."
At the same time, it's not bad in itself to use "he/she said." A great example of using it appropriately I think is actually in the first line of Bardrothen's second example set. There the "he said" acts as pseudo-punctuation. It breaks up the two sentences and I think gives each of them a touch more gravity, especially with how "It'll be a hundred dollars an hour" stands alone now. That's how you should view "he/she said," as pseudo-punctuation. You don't want to use a ton of them in rapid succession for the same reason you don't want to use a ton of short, one-clause sentences in a row, like "He opened the door. Then he walked out of his house. Then he looked both ways. Then he crossed the street." It's repetitive, and you have more tools at your disposal to control the pace and tone of your scene.
As for synonyms, I think where people really run into trouble with them is when the choice of synonym doesn't convey any information you couldn't infer from the dialogue itself. Here's an obvious example:
"Please, please don't hurt me," he pleaded.
We know he's pleading already, of course. And here's a particularly egregious example from an
actual bestselling novel. (Bonus points to whoever can guess which one it is!)
Aro started to laugh. "Ha, ha, ha," he chuckled.
Contrast this with something like this, which I think is an acceptable use of synonyms:
Kate was enraptured with my interesting story, but then something behind me caught her attention. "Hey, Jake!" she called out.
Here I'm using "called out" instead of "said" to convey with economy that Jake is somewhere in the distance and maybe hasn't noticed her yet. If I had used "said," the reader might assume instead that Jake is right behind me and was maybe already joining us without my knowing. Of course I could probably get a similar effect without a tag and describing a bit more action, like maybe she stood up and waved, but in this case I prefer the economy and that's okay.
I think the main takeaway should be that said-tags, synonym-tags, and no-tags-at-all are all normal parts of English writing and there's no absolute rule against using any of them, neither in grammar nor style. When evaluating a passage of dialogue, your reason against the choice of tags/omission of tags shouldn't be "Because 'said' is bad," or "Because synonyms are bad." Rather you should cite evidence of an undesirable
effect, like "Because it's repetitive," "Because it's overwrought," or "Because I can't tell who's talking." Figure out what works on a case-by-case basis.