Not immediately obvious but what I can't see is how you tell Google Ads that the conversion has been completed. The URL on that form does not change after completion (that was the random sumbission you just got!), so if I have understood your setup above you are probably seeing a conversion being reported at every visit to that page, rather than when the form is being successfully submitted.
My advice would be set a dedicated thank you page with it's own URL and add as a redirect on submit completion. Then you can update the trigger in GTM to not fire on all pages, but just on the page that is where the user gets redirected to after submitting. That should do what you want.
We tend to avoid Google Ads conversions preferring to set all the conversions via Google Analytics and then importing them into Google Ads, in this way the conversion data is more centralized. We do have some conversions recording in Google Ads (but not being counted) just in case we see some weird data in Google Analytics and want to cross-sanity check with what Google ads has recorded.
Hope this helps.
I would not wait for organic results, better instead to go and tell people with relevant websites - education institutions come to mind - about your great resource.
Consider now that in many verticals a first place organic result is going to bring you very little in terms of traffic due to the fact that paid search results are pushing all the free traffic off the page. That doesn't mean there are not opportunities, and it might be that in your own area of the keyword web - I've not checked - you still have the opportunity to get free traffic. I'm not saying you won't get free traffic, typically we see as much as 50% of all traffic as being organic, it's just trying to isolate those results, that are now heavily personalized by users (yes you can remove this when doing search analysis) so make it very difficult to actually see growth and measure against specific keywords.
I'd say almost you should ignore Google as you have a resource that lends itself very favorably to off-search types of promotion.
In answer to your specific question, if the website is brand new then you can expect to see a stabilizing of your pages in 1-2 months, if you are re-optimising existing pages then you might have to wait up to 6 for them to properly stabilize.
Games work well on Social Media, I'd be looking at Facebook or perhaps LinkedIn if you can afford it.