What document editor, word processor or layout software are you using to create your PDFs? Tagging Acrobat PDF works best with an integrated workflow utilizing MS Office, Adobe Indesign or Framemaker. Tags can be added at the time of document creation using these packages.
Some documents -- legacy, scanned or problem files will require manual tagging. I have seen many problems. But there are solutions and workarounds for most problems and they may not take long to do once you know how. Still tagging PDFs can be VERY frustrating at times.
Use the tagging panel in order to correct and define structure and reading order. When you use the reading order panel you may not be able to move one content item or element. You often need to move others as well in order to keep all content visible on the page. If something disappears, try moving other items in the content panel. Often the problem can be corrected by reordering other items on the page. If you are having problems with headers and other elements disappearing under background images you can delete background or banner images from the PDF and reinsert as backgrounds at the end, after everything else is tagged.
If you are selecting one element, and tagging it properly other items should not be grouped and tagged with it. I have never seen this happen. How are you selecting the item?
When you have a complex image that is causing problems (i.e., parts of it dropping out), create a simple image. One method: using the snapshot tool, select the image which copies it to the clipboard; create a PDF from clipboard; save the resulting PDF as an image (e.g., PNG). Use the edit tool to select and delete the original complex image from the PDF and import the simple PNG or JPG you created. Then retag the image. Alternately, you could create a version of the document with JPG or PNG and make your PDF from that. This will make your PDF more stable in general.
If your documents are huge and one page "goes bad" you can often make a tagged PDF of that page and insert in into your document, reorder/adjust the tags for that page in the tags panel and you will not have to start over. Do incremental saves as you work (e.g., XYZ01.pdf, XYZ.02.pdf, XYZ.03.pdf) so that if you have a problem you can rollback to the version just before the probem, or extract single pages.