Sigh.
This is going to be a long answer, but I hope it helps you make accessible PDFs from InDesign.
1. Upgrade to the latest versions of InDesign and Acrobat Pro. CC:2015 is quite out of date, and it's partly to blame for the mis-tagging you're getting in the PDF.
2. Yes, manual formatting (aka, overrides) causes <SPAN> tags in the PDF. The span tag is intended to identify that the text has been formatted differently...won't state the reason why. At this time, screen readers skip over span tags, but that won't be the case in the future.
3. Accessibility is not just for those who use screen readers: it's for people with sight, hearing, mobility, neurlogic, and cognitive disabilities. When you manually format text, you prevent many assistive technologies from adjusting it to what the user needs. Example: someone with dyslexia uses software that replaces the fonts in the PDF with dyslexie fonts which improves his ability to read the PDF. Or someone with low vision uses ZoomText which enlarges the text to 6-12" tall. Think of manual formatting as permanently hardwiring that font, size, color, etc. into the content and all of these assistive technologies can't override it.
Styles styles and styles. Make character styles when you can for bold, italics, underlining, or any other place where you would manually format text within a paragraph.
Create variations of your paragraph styles for what you need to do, such as adding more space above a paragraph or increasing/decreasing the indent.
4. Adobe is working toward the new PDF 2.0 and forthcoming PDF/UA-2 standards, so you're starting to see more tags in tables than before. Let them be. It's a bit clunky right now but that will be adjusted soon.
5. And in the thread Steve above referenced, that's a different problem. There's an error in the conversion from InDesign to PDF for the past 1.5 years where, if you view the PDF in the ORDER panel (not the tags panel), it shows every frigging line is wrapped in a <SPAN> tag. Doesn't appear in the TAG tree panel, which is what most assistive technologies are programmed to use. So ignore them for now as they don't appear to affect assistive technologies. Hopefully Adobe will correct this royal PITA soon.
—Bevi Chagnon
Member of the ISO committees for PDF and PDF/UA