The heatsink is a bunch of flap copper pipes. You dont need to replace those, but insuring they have a solid connection and thermal paste application to the chipset is something you should do.

Wouldnt want your repair foiled by an air gap.

The fan is something you should check on before you disassemble. Is it running at all? Is it blowing air? Is that air warm? A dead fan usually wont cause overheats alone, but it could be a factor.

Also, for the love of Mario; dont pay $21 plus shipping for thermal paste. That same tube of Arctic silver is only $7 on amazon.

https://www.amazon.com/Arctic-Silver-AS5-3-5G-Thermal-Paste/dp/B0087X728K/

Do the Trinity first, it's much easier than modding the Corona. My first Corona needed a post fix adapter and to bridge the missing resistors near the NAND header. It was much more work. Trinity is way less work. I would get the JR programmer because it can program Sonus 360 if you want to later on.

I would also like to suggest getting some magnifying glasses? The soldering points are tiny and these things make it so much easier.

I also bought a fiberglass pen to remove the thin coating on the soldering points. I don't know if you know how to take apart a slim, but the Xtools kit makes it much easier and I would also recommend getting an Xclamp removal tool. If you are doing the Trinity, hands down the Ace V3 with SRGH is the best. The V3 works great on the Corona too.

Some Coronas have a 16mb NAND that don't need an SD tool and then there's the 4GB NANDs that do. So you could potentially need an SD tool, a Postfix adapter and the Ace V3.I got my postfixs and SD Tool from PhenomMod. Weekendmodder has a video on how to tell if you need a postfix. It helped me a lot. I don't always agree with the things he says or his methods but he does have a lot of experience.

I have a list of the stuff I recommend. You can decide what you do and don't need.

Ace V3 Glitch chip: https://www.amazon.com/gp/product/B07SCZP2L8/

JR Programmer V2: https://www.amazon.com/gp/product/B01MTUWLVJ

Optional (makes your life easier or recommended for longevity)

Slim Xtools Opening Toolkit: https://www.amazon.com/gp/product/B07DNX4ZPZ/

Magnifying glasses: https://www.amazon.com/gp/product/B0827WBQYQ/

X-clamp removal tool: https://www.amazon.com/gp/product/B07BV42DSM/

Artic Silver Thermal paste: https://www.amazon.com/Arctic-Silver-AS5-3-5G-Thermal-Paste/dp/B0087X728K/

Header Pins (for flashing Ace V3, just easier, not required): https://www.amazon.com/gp/product/B07PKKY8BX

Fiberglass Pen: https://www.amazon.com/SE-7616SB-Fiberglass-Scratch-Brush/dp/B003NHDITW/

Yes, you might only need a screw driver and I would 100% buy some black thin zip ties at the $1 store near you. Oh shit now that you mention it you should absolutely get this https://www.amazon.com/gp/product/B0087X728K/ref=ox_sc_act_title_1?ie=UTF8&psc=1&smid=A1H9NMCPZH97BO
the stock thermal compound that comes with the CPU is normally garbage and this will lower your cpu temps by 10*. There is a reason ther are over 5000 5 star reviews on it. also if you dont have your own copy you will need to buy windows and barrow a disk drive to load it onto your computer. other then that all cables will come with parts. most will come with the PSU. Cables for connecting a monitor will come with the mobo and the gpu, some with the case. The case its self comes with fans already. you get the idea.