Aurora-R4, PSU replacement questions

Hello,

I want to first apologize about my poor hardware and general PC knowledge.

After having some issues with my computer, I figured out I needed to clean my case because there might be overheating issues.
As I dismounted the PSU, I found lumps of white paste (probably thermal ?) all over it. Although I didn't ask myself where it came from (I burnt my original graphic card, so maybe from there), I couldn't clean it without breaking the whole thing. It affected those "coils", which I presume are important, but also some small parts. I can provide pictures if needed.

I'm now looking for a replacement ; the built-in PSU is a 850W (if I remember correctly) and after some calculations, my station doesn't consume more than roughly 550 Watts (if my calculations are correct), but I have another problem.
Every modern PSU has an upper fan, compared to this one, which is lateral. I'm now asking myself three questions before buying something I wouldn't be able to use :

1. What is supposed to fit, format-wise ? I know this is a very dumb question, but I just want to be sure. I also do not know what I'm supposed to measure, and what safe distance I need to give for it to be optimal.
2. Will the connectors work ? There are some specific terms I don't understand, such as "ATX form factor" and some connector formats(I usually browse the products in french, so the translation might not be accurate), and I had to pull out a bunch of cables to get the PSU out. I'm wondering if I can plug everything back in.
3. Since I can't find one with a lateral fan (which I presume is the only one), will the cooling be affected, and thus leading me back to square one, with overheating issues ?

If it can help, I had the Corsair RM650i for reference, the 650W one.

I'm looking forward to your answer, and thank you for giving your time.

P.S: my components are the base Aurora R4 ones, I only added a hard-drive and replaced the AMD GPU with an nVidia GTX 1050Ti.

EDIT : I'm a mostly stupid for not checking beforehand, but it appears to be glue.