Are there any verification steps available to check if the installation of App and Vault server are perfect.
Kindly provide any suggestions
Answering your first question you can check that your Aras Innovator instance has been installed correctly by logging into your instance through the url you created during the installation. In order to confirm that the Vault server is setup you can create an item that will allow you to attach a document to such as a pdf or solid works part.
For example if you log onto your instance and go to the TOC > Documents and right click Documents it will give you the option to "Create New Document". Select this and give the document a name and add a file as a relationship such as a word doc or pdf file. Once you lock the item and it saves if you go to the specified location that you setup the Vault server to there will be a folder with the alias of your instance and within that folder there will be subfolders in which one will contain the same document you created within your instance. If you used the default location for the Vault it would be located in "C:\Aras\Vault\(name of your instance)" and the subfolders with the contents will be there as well.
Hope this helps!