I’d like to protect my content, main concern images, from being copies from my shopify site. I’m considering installing this ‘Cozy Antitheft’ shopify app, but I’d like to understand what disadvantages there would be by doing so? e.g. any SEO issues, particular browser quirks, google ranking/crawling/spider issues, Pinterest pinning issues etc (I’m not so concerned about the subjective ‘users won’t like it’ point).
First take other steps such as watermarking media content visibly, and non-visibly( ala steganography, papertowns, simple red pixel)
There are also DMCA takedown services that can monitor/search the internet for content reusage.
If your a business that sells to people you should always be concerned about the impact of such issues , especially if competitors don’t add user annoyance to their sites.
Not being able to right click and copy paste text, create a highlight link or share code, activate translation|spellcheck, crippling mouse gestures, or being unable to initiate a search on text is a big UX loss for users of modern browsers as casualties of trying to control the copy action.
If it seems subjective prove it and ab test any such change to be sure conversions are not impacted objectively.
It may not work - when javascript is disabled depending on the implementation, what an app is doing.
Some keyboards have context key, check if app works against those as well (if your demographic is in the graphics or technical space the occurrence of this keymay be higher than normal)
Accessibility is the biggest human issue, anyone with an assistive device or controls through right click menus will have a bad time unless they are technical enough to know to just disable javascript.
Google bots , SEO, really don’t care as they mostly interact with pages programmatically.
Biggest business issue is such things only stop casual right clicking by luddites, any bulk action will be programmatic not caring about any such apps, or it will be laughably bypassed by anyone with basic technical knowledge of the web or able to google search “right click not working”.
If you have a niche product where the imagery itself is the product and are selling in a type of ecosystem where plagiarism, sharing, copycatting, etc is common among your audience then sure you should add barriers against that behavior.
However if imagery is the only product you have then seriously need to be working with a DMCA takedown service and registering copyrights instead of relying on default copyright protection at creation or whether right click is actually being controlled or not.
Thanks @PaulNewton for your thorough answer 
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