How can we check if the SDK is properly initialized?

You can see the Insider One session start response once the application is switched to the foreground. See below for an example of  I/[INSIDER][sessionStartResponse]:

{
  "analytics_status": true,
  "sdk_disabled": false,
  "social_proof_enabled": true,
  "inapps": [],
  "contents": [],
  "passive_variables": [
    "loginIsHidden",
    "nawButtonText",
    "contentOptTest9",
    "loginText",
    "endpointString",
    "bool_variable_name",
    "quickview_banner_2021-02-08-Móveis",
    "contrast",
    "itemBackground",
    "mainColor",
    "settingsButtonText",
    "intGet",
    "mob-6046",
    "int_variable_name",
    "loginTestCount",
    "purchaseButtonText",
    "button1",
    "string_variable_name",
    "ns_string",
    "stringGet"
  ],
  "smart_recommendations": {
    "1": "https://recommendation.api.insiderone.com/shopbagg/{language}/most/purchased/product?category={category}&subCategory={subCategory}&details=true&platform=mobileV2&size=25",
    "2": "https://recommendation.api.insiderone.com/shopbagg/{language}/complementary/product/{productID}?currency={currency}&details=false&platform=mobile&size=10",
    "10": "https://recommendation.api.insiderone.com/shopbagg/{language}/most/purchased/partner?currency={currency}&details=false&platform=mobileV2&size=10"
  }
}

Check the [INSIDER][sessionStartResponse] string inside the console of your IDE to see if the Insider session starts properly.