I can edit, more or less, but when I click on post-editing buttons like Upload/Insert Media, or (over on the right) try to edit to edit the "published On:" date, nothing happens.

If you are trying to create a post on the WordPress.com home page here http://wordpress.com/#!/post/ then forget that please. Instead create posts only on your own dashboard here > Dashboard > Posts > All Posts > Add New
http://chickadeeworkshop.wordpress.com/wp-admin/post-new.php

For details about the options for uploading images I recommend
http://wpbtips.wordpress.com/2009/04/26/images-the-three-link-options/

For troubleshooting images tips please see > http://en.support.wordpress.com/images/troubleshooting-images/

Keep us posted re: the links issue.

(2) Exactly which browser and version of it are you using. If you aren’t sure click here to find out > http://whatsmyuseragent.com/

Note 1: If you are using an IE9 browser in compatibility mode that makes it render as IE7 and IE7 is not supported. To disable see > http://www.sevenforums.com/tutorials/1196-internet-explorer-compatibility-view-turn-off.html

a. Try clearing your browser cache and cookies > http://www.google.com/support/accounts/bin/answer.py?hl=en&answer=32050

b. Try using the browser with all extensions or add-ons disabled

c. If applicable try updating the browser to the latest stable version http://browsehappy.com/

d. Make sure third party cookies are enabled on the browser http://support.google.com/accounts/bin/answer.py?hl=en&answer=61416

e. Also confirm JavaScript is enabled http://www.google.com/adsense/support/bin/answer.py?answer=12654

f. Then confirm Flash is updated http://www.adobe.com/software/flash/about/ and if required installing the latest version of Flash http://www.adobe.com/products/flashplayer/

g. Try resetting your router and modem and rebooting your computer and then logging in using the secure https:// log-in

h. Try using another browser. You can download one here http://browsehappy.com