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To embed a YouTube video in Squarespace, follow these steps:
Go to your YouTube video and click on the "Share" button below the video player. Click on the "Embed" option, and you will see the embed code for the video. Copy the embed code. Go to your Squarespace website and navigate to the page where you want to embed the video. Add a new content block by clicking on the "+" icon in the top left corner of the page. Select the "Embed" option from the content block options. Paste the YouTube video embed code into the content block. Save the changes to the page. Your YouTube video should now be embedded on your Squarespace website.
 
If the embed code isn't showing up, make sure the video is shared to the public.

Trunk Inventory is the only one that I'm aware of. The good news is that you can try it out for 14 days. The bad news is that it's not cheap at $35+ per month (depending on order volumes).
There are other players in the inventory sync space but, to my knowledge, none of them sync with Squarespace yet.
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Unfortunately, it is not possible to completely prevent someone from sharing a YouTube video that is embedded on your Squarespace website. When you embed a YouTube video on your website, the video is still hosted on YouTube's servers, and YouTube provides the ability for viewers to share the video through social media or by copying the video's URL.
However, there are some measures you can take to make it more difficult for someone to share the video outside of your website. One option is to disable the "Share" button on the video player, which can be done within the YouTube video settings. Another option is to set the video to "Unlisted" on YouTube, which means that it will not appear in search results or on your YouTube channel, but can still be accessed through the direct link.
Additionally, you can add a message or disclaimer on your website asking viewers not to share the video outside of the page, or consider password protecting the page where the video is embedded. These measures may not completely prevent sharing, but can help to discourage it.

Glad that working now!

It appears that the problem was that Squarespace didn't detect a full disconnect from the previous Squarespace site because the nameservers on GoDaddy were still pointing at Squarespace. The resolution involved reverting the nameservers back to the GoDaddy default, and then switching them back to the Squarespace nameservers.

The screenshot of the Squarespace domains page doesn't look like you have set that up with nameservers hosted on Squarespace, it looks like standard manual DNS setup. You may want to disconnect the domain on Squarespace and go through the setup again and make sure to go the nameserver route.

Looks like the hamburger menu is taking up 47px of space pushing the logo that far to the left. If you add a -47 right margin to the logo it should center (see screenshot).