Most of the time, yes they affect your load speed.
You can run the page through webpagetest.org to get a "waterfall view", this will show you every file that loads on the page, how long it takes, file size, etc.
If your videos are below the page fold (so you have to scroll to see them), you can lazy load them, so they don't show up until the user scrolls them into view. Here's an article I wrote about lazy loading, it applies to images and videos: https://speedboostr.com/shopify-lazy-loading.
To answer your questions specifically:
Since I don't have any video on my homepage, page speed would only be a concern or the product pages the video is on, correct?
That is correct, the assets from your product page do not load on home page (other than shared scripts but not page content).
What is the maximum size a video should have?
The best thing you can do from a performance perspective is put them "under the fold" and lazy load them. Then they won't contribute to page load speed. For above the fold, you want them to be the smallest possible size possible while retaining an acceptable amount of resolution. Videos are heavy so minimizing size is a best practice.
Founder of
Speed Boostr (Shopify optimization and dev agency).
Creator of
Theme Scientist (A/B testing app).
Creator of
Order Automator (app that auto tags + fulfills orders, send emails).
Creator of
Tip Jar (add a tip button to your store).
Creator of
Shopify Analyzer (free performance analysis tool).
My team also builds custom apps and automated solutions.
Enjoy the adventure!
