Looks like the drive experiences overheating issue. Unfortunately it's a common issue for Seagate drives which happens with time (had exactly the same problem with iMac 2010 21.5`` and stock 500 GB hdd). I used free Macs Fan Control tool to cool down the HDD (specially in Boot Camp, where the drive was even hotter). The article is about HDD replacement but it also explains how to use the app.

But the best solution I think is to replace the drive with some 2.5`` drive. They are known to be much cooler and quiter (though slower)

SMC Reset

To your minor question: I do not think the absence of the battery should affect the SMC reset. Source: I have successfully done this with an older mac with no battery and corrected fan problems.

Key Observations

Since it is not obvious what your problem is, I will cover some bases starting with some key observations:

1. You did not mention resetting the NVRAM/PRAM which I also did when fixing my fan issue. Have you also reset the PRAM/NVRAM? That would one of the two immediate troubleshooting steps I would do with something that looks like a firmware issue. (I believe you hold down the key for 3 chimes btw… not sure if it tells you that on the site.)

2. Since your diagnostics do not appear to be reading your CPU temperature accurately (it should definitely not be 0C (34 degrees F). I would try again with a different diagnostic, and/or repair permissions using diskutil, run with sudo to ensure the program is able to access the requisite files to generate real output.

Troubleshooting

Some possibilities and thoughts to facilitate further troubleshooting:

While in diskutil I would check that your drive is in good health. HDD's only operate for 5-10 years. My 08 pro has a dead drive… it may be working hard to keep it at a low temp so it doesn't lock up? Physical wear on the drive can also generate heat.

I would check out your system diagnostics to see if anything is hogging resources. Assuming your hardware isn't screwed up and it's not firmware, I would check to make sure the kernel isn't panicked or some weird unrecognized process isn't interfering with your system.

I note that without an intermediary battery your computer may not be getting a completely regular power source… what comes out of your wall may be lower or higher voltage than expected. Sometimes that generates heat and causes fans to go insane. Sometimes that can "zap" files. You may wish to reinstall your OS without deleting your files and configurations, to make sure everything is in good order.

You may also wish to get your hands on a battery to see if that makes a difference. These cannot be too expensive nowadays… someone may have one laying around. The fact that yours once exploded also… you may be able to get one from Apple even if you're not covered. They really don't get a lot out of keeping and warehousing old parts (why applecare is a good money maker for them) and like happy campers and I think that battery was recalled no? You may have been entitled to a new one years ago anyway.

You may also have already tried this but checking the logs at /var/log/system.logs (hard to parse) may provide a big picture view of what's going on with the fans even if it's not entirely understandable or easy to get through… sometimes the error messages are verbose and incredibly precise. Hit or miss. It will be FULL of entries that don't matter, but may tell you quite clearly what's going on if you just do a find for like "fan" "heat" "power" "voltage."

Finally (albeit obviously) you may want to make sure you don't have a ton of crud clogging up your fans or blocking. Cleaning a computer directly or with air can easily mess things up… but yeah… this may be necessary. If a sensor cannot get a read on heat or a fan is covered in gunk it may run the fans at full just to get normal cooling in order (first case) or because it's designed to be cautious about the heat issue (second case).