normal.dot from 2003 to 2010 will not accept

It sounds as though you have successfully copied the macros etc from the old 2003 template to the new one, and that you have modified the new template with ribbon commands etc. That being the case you should be able to copy that modified template to the other PCs and the contents will go with it. There may be some anomalies as we don't know exactly what is in the template.

You should also be able to rename that modified Normal.dotm template (to say MyMacros.dotm) and use that as an add-in on the other PCs, by copying it to the Word startup folder. To what extent your new ribbon commands will work depends on what exactly they do.

If you have not changed the preferred startup folder it can be located (in English language versions of Windows) by typing
%appdata%\Microsoft\Word\Startup
in the Windows Explorer Address bar and pressing Enter.

To repeat from my first post:
1) Normal.dotm in Word 2010, not normal.dot.
2) Do NOT rename normal.dot to be normal.dotm
Templates in Microsoft Word
Normal Template in Microsoft Word - How to Open

Normal.dotm should not be shared. You should not take the normal.dotm from one computer to use as the normal template on another computer. The normal.dotm for Word 2007 is different from that for Word 2010 is different from Word 2013.

Temporary measure:
Put normalold.dot in the Word Startup Folder. You will have access to all macros and AutoText. Do not start building your ribbon customizations on that basis.

Your lost new AutoText entries.
These may be in your renamed normal.dotm file. They may be in the Building Blocks folder. Check it out by opening the file directly and looking for the AutoText.

How to share macros and AutoText with multiple computers

You can create one or more Global Templates to share macros and AutoText and other Building Blocks. These can also hold QAT modifications but not ribbon modifications that you make from within Word.  Modifying the QAT and Ribbon in Microsoft Word

You can make Ribbon modifications that can be easily and flexibly shared in templates through :XML and vba but that takes more work.  Customize the Ribbon (It doesn't take rocket science)

AutoText must be stored in a template. That template, if you want the AutoText entry to be available to all documents, must be made global by putting it in the Word Startup Folder or in the Building Blocks folder. If the template will also hold QAT modifcations and/or macros and/or keyboard shortcuts, it must be in the Word Startup Folder. Building Blocks are, in many ways, AutoText on steroids, but AutoText is one kind of Building Block in the Ribbon versions of Word.
Automated Boilerplate Using Microsoft Word
Building Blocks & AutoText

Don't forget your AutoCorrects if you built a library of those.

Macros must be stored in .dotm or .docm files. Since you want yours available in multple documents you will want them in templates, and probably in Global Templates.

Within the .dotm or .docm file, macros are stored in modules. If they are recorded macros, they are in the NewMacros module. They can be stored in any macro-enabled template or document. They do not need to be in your normal.dotm template but can be there. For purposes of sharing them, you will want them in a different template, a Global Template.

First: You want to see the NewMacros module in your normalold.dot. To do this. open the file normalold.dot so that it is on your screen. Do not type anything!  Press Alt+F11. This gives you the Visual Basic Editor. If you do not see the "Project Explorer" on the Right, press Ctrl+R.



You should see two "projects" named Normal:
Normal
Normal (Normalold).

You want the second one. If you do not see the NewMacros module under it, click on the little + box in front of the project name.

All-at-once method
Right-click on NewMacros and choose to export it. The suggested name will be NewMacros.bas. Change it to something like modCKK where the last part is your initials. Save that on the desktop.

Then create a new dotm template to hold your macros. Go into the Visual Basic Editor (VBE) and find the project for that document. It will be named TemplateProject as a default, but we should change that. If a Properties window shows under the Project Explorer, look there when you click on the TemplateProject. If the Properties window is not showing, tell the VBE you want to see it by picking that option under the View menu. Then click on the TemplateProject in the Explorer.
(wish I could upload screenshot here)

In the Properties window it is likely that the first (or only) property will be name and it will say "TemplateProject." Change this to something short that you will recognize. Your initials will work.

Once you've renamed it right-click on it in the Project Explorer and Import your module.

This template now has all of the macros from your normal.old file. You can put it into the Word Startup folder and the macros will be available in all of your documents. You can move your normalold.dot file out of the Startup folder now. Keeping it there will cause confusion and needless system overhead.

You can store keyboard shortcuts and QAT modifications that access those macros in your new template.

More organization than all-at-once
You may want to separate your macros into more than one module. You can use the vba editor for this and actually look at your macros. You can cut and paste individual macros from one module to another. If you are going to do this, you want to do it before you do anything with keyboard shortcuts or QAT entries that access the macros.

Ribbon Customization
Ribbon customization from within Word is really only designed for a single computer. You can export and import these customizations but it is an all-or-none process. If you import customizations you overwrite all customizations already existing. This is not good, generally.

For sharable flexible ribbon modification, see Customize the Ribbon (It doesn't take rocket science).

See also Distributing macros to other users.