There are the heading styles, which can be modified to have numbering. See How to create numbered headings or outline numbering in Ribbon Versions of Word by Shauna Kelly.

The heading styles will be reflected, by default, in the TOC dialog. Generally only the first three are included by default. When you click OK in that dialog, it creates or modifies a TOC field. Which styles are included are set in the TOC options. Whether those styles have numbering is a function best controlled through the process set forth by Shauna Kelly's article. If you change the numbering in the underlying style, it will change in both the TOC dialog and the actual TOC.

How the individual levels in a TOC appear is controlled by the TOC Styles. There are nine of these, one for each level. Numbering is picked up from the underlying heading styles. The TOC styles, themselves should seldom, if ever, have numbering applied to them. The TOC styles are not normally in the [Quick] Styles Gallery on the Home Tab; the Heading Styles are in that Gallery. You can access the TOC styles directly in the Styles Pane.

• The Styles Pane by Suzanne Barnhill, MVP
• Modifying TOC Styles

The appearance of the individual entries in a TOC can also be changed by direct formatting applied in a heading. See Effect of Direct Formatting by Suzanne Barnhill.

If you select a TOC in a document, you can save it to your TOC Gallery. That saves which styles are used and the other options set in the Custom TOC dialog. It does not save the underlying heading styles nor the TOC styles.

If you copy a TOC field into a document, it will carry the Option settings from when that TOC was created. This is true whether you copy and paste from a document or whether you choose it from the entry you saved in the TOC Gallery. Add your own Table of Contents to the TOC Gallery

A TOC field can also be edited manually by displaying the field codes rather than the result in the document.

• Dealing with Fields in Microsoft Word
• TOC Tips and Tricks by Suzanne Barnhill, MVP
• TOC (Table of Contents) field

By default, since they are in the [Ouick] Styles Gallery, heading styles will be included in a [Quick] Style Set. If they are numbered using the methods in Shauna Kelly's page, my experience is that the numbering scheme will also be in a Quick Style Set even though Numbering List Styles are not something that can be saved. You can tell Word to add the TOC styles to the [Quick] Styles Gallery on the Home Tab. If you do that, they will also be included in a [Quick] Style Set created from your document, which can then be applied in a different document.