How to promote/open app from Google search website results?

For your first question, what you might be looking for is what is called App Indexing. As Google itself describes it:

App Indexing lets Google index apps just like websites. Deep links to your Android app appear in Google Search results, letting users get to your native mobile experience quickly, landing exactly on the right content within your app.

Basically, the App Indexing API is the one responsible for informing the web of the deep links present in your application.So, first of all, you'll have to add those deeplinks to your mobile application. You can do that in android by specifying Intent Filters and defining a schema like described here and on iOS by defining URLTypes as described here.

Once you have your deeplinks working, you'll add the App Indexing API to your app through Google Play Services.Basically, you'll add calls to every onStart() and onStop() of the Activitys you're interested into indexing with code like:

// in onStart()AppIndex.AppIndexApi.start(mClient, viewAction);// in onStop()AppIndex.AppIndexApi.end(mClient, viewAction);

You can also find more details for the API and how to set it up in detail here.

Since last month, you can also drive installs of your app through App Indexing, as noticed here.

PS: If you're interested, there are nice DevBytes videos about App Indexing here and here.

The search box in Google Chrome results is called Sitelinks Search Box, so if you do have already a querying mechanism on your website, all you have to do is implement the schema.org markup on it:

<script type="application/ld+json">{   "@context": "http://schema.org",   "@type": "WebSite",   "url": "https://www.example-petstore.com/",   "potentialAction": {       "@type": "SearchAction",       "target": "https://query.example-petstore.com/search?q={search_term_string}",       "query-input": "required name=search_term_string"    }}</script>

After that, you'll have to wait a little for Google to update its crawlers with your new information. You can find detailed information about the markup with examples and troubleshooting here.