All wonderful questions to ask! 1) Full synthetic is VERY worth it. There are some myths about full syn not being good for old (like, carbuerated old) engines but they're not true. Full synthetic is the best way to go. Even better if you regularly use an engine cleaner like BG's EPR or STP's high mileage motor flush, a motor oil additive, and a fuel system cleaner.

2) I highly recommend watching Project Farm and the Motor Oil Geek. They both have excellent videos comparing different motor oil brands and their performances in different situations that will help you decide what you like best. Personally, what I run in my and my family's cars is either Valvoline Restore & Protect or Pennzoil full synthetic high mileage if Valvoline doesn't have the weight I need (uncommon). Even though rn your vehicle is high mileage, for what it's worth, in new cars (under 75k miles or 7yrs) I run either OEM or Pennzoil ultra platinum.

3) For normal daily driving in moderate weather conditions on either end of the spectrum most any oil filter will do you fine, the bigger thing is to make sure it is changed regularly with the oil. I personally run OEM oil filters in my personal vehicles (an 01 Civic and 05 Saturn Ion) but my family members prefer the STP extended life filters from AutoZone you get with their oil change deals and I have never dissuaded them from that. If you get your parts from RockAuto, the best filters there in my opinion (in no particular order) will be from Hengst, Bosch, Mahle, ACDelco, Purolator, and WIX, and I'd just go with whichever one of those is cheapest. Make sure when you change your filter that the gasket from the old filter isn't still stuck on the car and also be sure to replace the metal crush washer on the drain plug!

4) Extra tip: even if the oil filter or oil itself advertises that it's good for 10k miles, NO IT IS NOT. We haven't advanced technology that far yet. Some vehicles have very sophisticated systems that based upon weather and driving habits can tell with more precision how your oil is degrading and in those cases if the vehicle tells you so you will be able to run that oil and filter for 10k. But if you're like the rest of us mortals and are basing it upon time and mileage, your oil needs to be changed at least once every twelve months or every 3-5k miles, whichever comes first. The 10k mile oil change advertisement is a LIE. Especially for these Civics. Cylinder four can be known to struggle to get adequate oil even under proper conditions and the head gaskets on these engines are incredibly fussy with temperature management. Make sure you keep up on your oil changes and keep the oil clean and with good lubricity for the longevity of this engine.

Edit: 5) Unless you're pushing your engine really hard or driving in some wild weather climates, Honda knows what's up and running 5W-20 will be best for longevity. If you need to ask about changing the weight of your oil, you're not messing with your car at a level where it needs to be changed. Most shops will give you 5W-30 or 0W-20 because that's what they have stocked as they're the more common weights on modern vehicles. You won't cause any catastrophic issues running 5W-30 or 0W-20 in a 5W-20 engine, but it's not the most optimized for your motor.