Yes Google understands H2s as a subtitle and this helps with your page's readability and optimization. I would not remove them, and instead see if there are any opportunities to optimize them with target keywords or LSI keywords.

According to Ahrefs, 21% of their referring domains have the exact anchor text "meetoo", this is leading Google to think it that https://www.vevox.com is relevant for that term. Also the old website meetoo.com is 301 redirecting too vevox.com, which is another signal that Google is seeing that also makes them relevant for "meetoo".

I would recommend keeping the 301 redirect in place, but you may have to do a "link audit" to see if you can edit some links pointing to https://www.vevox.com that have the anchor text "metoo" as well as build or gain new links that say "Vevox" or "https://www.vevox.com" to help outweigh them.

The other option is to wait it out as I imagine Google will eventually figure out that https://www.vevox.com is not what the searcher is looking for when they search "metoo", unfortunately the wait it out method could take months or longer.

I would check the Index Coverage report in GSC, and then maybe try re-submitting your sitemap in GSC.

For Google Analytics I would check under the Source/Medium report to see if Organic Google traffic has changed at all.

Someone may have a workaround but my understanding is no unfortunately. Google Analytics does not support pulling the zip codes or geocoordinates in any way. In addition, you should not send the data as custom dimension to GA as it is against the terms of service:

“If collecting geolocation info, ensure it is not GPS or fine-grained location information, as this could lead to reasonable inference of the individual. “Fine-grained location” information for Analytics is defined as any area less than 1 square mile, including any lat/long data. In some instances, such as in the UK, zip code can map to a single residence and thus cannot be passed to Analytics.”

https://support.google.com/analytics/answer/6366371?hl=en#geolocation

I believe there is a way to do a "Find & Replace" function within Wordpress or your HT Access. I would highly recommend working with an experienced website developer for this. If you have any additional Schema in the of your website, updating it within the should update it sitewide automatically.

Sometimes it just takes a couple days to show in SERP results. I looked up the listing and it appears there are now pictures showing on your GMB listing.

Not that I know of unfortunately.

If they were local business listings you could use Moz Local, Yext or BrightLocal, but it doesn't look like that is the case. I would just let the client know it may take some time, but with continued link building efforts Google will eventually start to drop Vevox from the search results for MeeToo.