I'm only going to address part of the question here.
Please see the above image.
In that picture, my Weebly blog is open for editing, but I have not yet begun to edit a specific post.
The “Blog Sidebar” tab is active in the left-hand menu.
The key point is that you need the specific HTML code. For example, if you have “blogroll” code in a different blog, you could copy-and-paste that into an HTML area. Offhand, I don't know how to code a blogroll.
It won't work if it's normal Weebly text, because that forbids HTML codes...if memory serves.
Of course, Google or Google-Plus or StumbleUpon give their members the sharing code from their logged-in sites.
Once you have the code in place, you have to click the "Publish>>" button in the upper-right of the top menu (where it says "Publish>> ? Help! X Close").
Recap for Adding Widgets to the Weebly Sidebar
- Edit the blog, but do not select any one article. (In the example shown, I was not editing any one blog post. Now that I'm editing this one, the "Sidebar" option is not available).
- Drag and drop a widget into the sidebar. Just now, I don't have time to figure out which type(s) of widgets permit (HTML) code rather than normal text. Try it and see...and leave a comment if you figure it out before I have a chance to do so!
- If Weebly supplies the right widget, just skip the next point; you don't need to change the code.
- If Weebly does not supply the exact widget you need, you probably need to get it from another reliable source. I know that's how I got the G+ and StumbleUpon buttons.
- Once you have the code in place, click your mouse in a different widget to finish the current one.
- Once you're done, click the "Publish>>" button in the upper-right of the top menu line. (That's the row with "Publish/?/Help!/X_Close").
- Try it out in "reader" mode.
Updated on Jan. 22, 2013
Second, and more importantly, the "Custom HTML" widget is found in the Elements + More + Misc menu, but only when updating a blog post (or possibly a static page).