Google Remarketing Tag

Generally If I were you I would take this free course of Google first:
https://analytics.google.com/analytics/academy/course/5
This helps you understand the basic concepts with too much technical details.

I am getting confused about where I need to be adding the Google Conversion Tracking Code and Google Remarketing Codes. (assuming not doing any dynamic ads)

Do I need to add the Google Adwords conversion tracking code to my check out success page manually and then also setup a Trigger in GTM on this page “checkout/order-received” ?

Usually the Google Remarketing tag is fired using the “All pages” trigger within Google Tag Manager. This will allow you to build target audiences based on every site visitor wherever they land on your site.
The conversion code should be only fired on the page that you treat as your conversion page. This can be a successful newsletter subscription thank you page, a white paper download or in your case: an order properly placed in your webshop. You do not need to add the code manually into your site, you just need to add a new tag into your GTM container with the tag type Google Ads Conversion and enter your conversion ID provided by Google Ads.

This new tag should have a new trigger as well which will have a type of “Page view” and while setting up this new trigger, you will need to select the “Some page views” option and then select the Page URL variable as a parameter, use the “contains” relation and enter “order-received” into the variable value fields. This new trigger will make sure your Google Ads conversion tag will fire only on the order received page.

If you are using Google Ads conversion codes to measure success within Google Ads, also add the Conversion Linker tag into your container to make sure more restrictive browsers like Safari will also give you better measures.

This is so confusing, I understand that Google Adwords has special conversion tracking (but I don’t really want to use that if I can just use Analytics goals, and I don’t want to duplicate conversion data?), that said I understand that Google Analytics has “goals” for conversion tracking which can be imported into Google Adwords (this is my preferred method), but I am unclear on this point about adding to GTM and how the newer advancements in 2019 affect things.

Next to GA goals, you can also import ecommerce transactions from Google Analytics into Google Ads and my advise would be to do so instead of using Google Ads tags. If you enable Ecommerce tracking in GA’s View settings pane (https://support.google.com/analytics/answer/6032539?hl=en) and you have connected this GA view to your Google Ads account, you can import transactions just like goals on the Google Ads UI.
https://support.google.com/analytics/answer/1034306?hl=en

Google Tag Manager adds another layer of confusion about whether or not it’s important to remove the hard coded Google Adwords conversion tracking scripts from the actual success pages themselves, or if setting up the triggers to fire on those pages only inside of GTM is sufficient?

You should always de-duplicate your tracking codes if you fire a tag using GTM. If you do not remove hard coded codes while also using them in your GTM container, you will end up measuring the same amount and number of actions multiple times.

I hope this will help you better understand how GTM could help you.