It could be the battery - but as I said before, it's my best guess but one that I wouldn't bet a lot of money on. However, the battery almost certainly needs replacement anyway.
I don't think Apple will ship the battery to you.
https://support.apple.com/kb/index?page=servicefaq&geo=Canada&product=Macnotebooks
(The search results had Canada selected - just select your country.)
It's doesn't hurt to do the SMC reset again - I've seen the several times thing but I don't know if that's just OCD or not.
A different browser can make a difference in this way: different browsers will take differing levels of CPU power to process the same web page. I did a test a month or so ago and found that when viewing a YouTube video on my 2012 Mini, it took quite a bit more CPU to run in Firefox than in Safari. It also will vary depending on the graphics hardware you have. With the Intel HD4000, which is the GPU that I have, there are sometimes problems in viewing some YouTube videos on Firefox, sometimes this can be avoided if I don't use hardware acceleration. But then if I don't use hardware acceleration, it uses even more CPU power. These YouTube videos usually run problem-free in Safari with hardware acceleration. You can try and view the same video in the different browsers while seeing what kind of CPU usage and temperatures you have. If there's a big difference, then maybe it'll help if you use Safari (or whatever browser is more efficient) for those pages. I'm pretty certain Safari will turn out to be more efficient. I don't use Chrome but people say it's a resource hog.
HTML5 is a specification for the markup language used for web pages. In web video terms for the end-user, browsers such as Safari and Firefox support it when you install it, but for something like Adobe Flash, you have to install the plug-in to view it.
EDIT: Your latest post came as I was writing this. Hopefully you found the solution to your fan issue. But yes - if you want to continue to use the computer for a few more years, then you should replace the battery.
Treekram, thanks for all your thouthful posts. Once I fix this (tomorrow), i'll come back here to confirm that my theory was right...or I'll be seraching for other issues. Regarding Safari...and what you've written. I've been using firefox because of one plug in that is really important to me (it's a software that helps me read Japanese written languge). And I got so used to that browser that it's harder to make a switch. But would you say that Safair overall might be the smoother and better performer for Macs given it's one of the integrated apps with OS X?
also, still have a question regarding the battery: you say it needs replacement. But using it now, it shouldn't hurt my compuer right(that's also what the message says when I click on it:
"Service Battery: The battery isn’t functioning normally, and you may or may not notice a change in its behavior or the amount of charge it holds. Take your computer in for service. You can continue to use your battery before it’s checked without harming your computer."
the issue is that I really don't have access to any Apple service..but will be in couple of months in DC and there I can do it easily. Meanwhile I'd keep using this one (or I just found a vendor on-line here who sells this...but they don't seem to be an authorized dealer, though a shop that deals with all things by Apple..they do sell this battery, would that be a legit buy?)