There's no way to know the problem from your description. You'd need to look for patterns in the way the page is coded. Extensions could possibly be the culprit, but they usually aren't, despite the fact that they're by far the most widely suspected. Whenever there's any browser problem, half of the respondents will tell you to disable your extensions.
1) Think twice before updating software to "bleeding edge" versions. If you
*must* use risky software like Flash it should be kept updated. Otherwise, in most cases you're just working as an unpaid beta tester by getting the latest when it's released, usually for no good reason.
In the case of Firefox, Mozilla has been breaking it by bits and pieces, removing user-friendly functionality to satisfy their boss, Google. (Yahoo has recently contracted to provide default search in FF, but in the past nearly all of Mozilla's $100+ million/year budget has been coming from ad company and privacy enemy Google.) So updating FF is a crapshoot. There's no telling what might have been broken or foisted onto people in their latest remake.
2) Search online for other reports of similar problems, such as "rollovers not working Firefox 35". Someone might have already done the work you're now faced with.
3) My first guess would be javascript settings. My second guess would be the use of unsupported CSS in the webpages. But guess #2 is very unlikely in the latest FF. Unfortunately, newer versions of FF have actually hidden the javascript settings! You can replace them with an add-on:
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Then try going to Options -> Content and click the Advanced button next to the javascript option. There are some options there. But what you'll really need to check is the user_prefs, which is an abstruse mess. If you're feeling ambitious, enter about:config in the address bar and click Enter. In the search field enter "dom" (without quotes). Dom stands for document object model. The settings that start with dom relate to what javascript can do. A rollover can be done with CSS, but that's a recent ability and requires a web designer who knows what they're doing. The vast majority of rollover effects use a very old javascript trick. Unfortunately, it's not easy to understand what you're looking at in about:config. I don't see anything specific to rollovers. But you might check that this setting is set to false: dom.disable_image_src_set
The calendar problem could be related to script settings related to opening new windows, but that's just a wild guess. There's no way for anyone to know what's not working with your AT&T webmail. First, most or all of us don't use AT&T. :) Even if we did, you didn't describe what doesn't work. (Have you considered getting your email through a real email program? It's much better functionality than webmail.)