I think the idea with the BHA first, and AHA after (if you feel that you need both types of chem exfoliance), is that the BHA gets into the pores, while the AHA deals with the top layer, I think the idea is that the BHA allows the AHA to work a bit better, but I'm not totally clear as to why. Honestly, if you haven't used either a BHA or an AHA before, I wouldn't just jump into them both at the same time or more than once a day. I did, and it irritated the shit out of my skin. I'm able to use them both now, but it took a few months for my skin to get used to it and I had to introduce them one at a time. Even today, I typically use one for AM, and the other for PM, and not everyday.

That sunscreen is fine as its spf is over 25 and it is broad-spectrum, but make sure you use a proper amount of it (1/4 tsp is usually what's recommended around here, which is quite a lot and may change your opinion on the product you decide on). Personally, I think moisturizing is a very important step, for oily and dry skin types, and I personally have never found a sunscreen (even with moisturizing properties) to give me that moisturizing punch I need. I would recommend buying a sample or trial size of the sunscreen (Paula's Choice is cool like that), as sunscreens are super hard to settle on one that works for you, and just double check to see if it feels moisturizing enough for you.

Paula's Choice is an excellent brand with a pretty substantial cult following (not just on this sub!), and several products in particular are pretty big hits. Not all of her products are that great though, so I definitely recommend searching through this sub for reviews (you'll find hundreds, but maybe not as much for her sunscreens as it seems that this sub favours Asian brand sunscreens). Makeupalley.com and r/AsianBeauty are also great resources in getting reviews for sunscreens.

Routines are hard to recommend, as some people find that using lots of products help, while others find that doing the bare minimum helps them the most. A popular expression around here is "Your Mileage May Vary" (YMMV). Some people like to just splash their face with water in the AM (followed by an optional exfoliant, moisturizer and sunscreen), and a cleanser + moisturizer in the PM. Some people find the oil cleansing method (often just in the PM as a sunscreen/makeup remover), usually followed by a cleanser, to be more gentle. You could definitely use a cleanser in the morning if you think you need it, but some find that to be over-cleansing and prefer a splash of water. I use a thick moisturizer and vaseline at night for my dry skin, so I prefer to cleanse to get that off in the morning.

The routine you're thinking of sounds fine (if your skin is ok with it), though I would recommend cleansing in the PM to get rid of the sunscreen, following by a moisturizer. If you want to save the chemical exfoliant for nighttime before moisturizing, that's fine as well, as I think the AHA needs 10-15 mins on its own before something is put over it (I'm not positive, people go back and forth about this on the sub). Lots of people around here love OCM or the double cleanse to remove sunscreen, but there are a lot of products that can do the job. Developing a routine is really about trying on products one at a time (so they don't irritate your skin, and if one is problematic, you know which one it is), making sure that you patch test, and seeing how your skin handles them. It's slow and can get kind of expensive :(

For your last question (not stupid at all!), you should only wash your face at the start of the routine.

I hope this helps somewhat!!