Using a header and footer for design layout is very normal. A very good practice that allows you to update one page to change every page that includes the file. This is a very basic good design practice. NOTE - I always have everything from the opening html tag, all the head (therefore all the included CSS and JS
files) and body tags and usually all the outer div tags (or tables if its using tables) in my header file. The only content in the actual page is to place the content and any content/page specific code needed. The in my footer I will have the closing outer div or table tags, closing body/html tags
;;;they also create a layout one by one in every web pages they have for displaying a dynamic content.
I don't understand what you are saying here. The header and footer should be included in every page. This is where the design should live. The page contents should be just the content. Sometimes this content may have some design but this is content/page specific.
;;;how can i change this website's theme/design/style to make it look like brand new website?
You either have front end design skills or you don't. It looks like you dont or you would know the answer to this question. The answer is you write the HTML/CSS/jQuery to create the design. You may have to engage a front end developer to build this or use a library e.g. Bootstrap. This is a CSS framework which can provide you an easy way to create a responsive modern design . Google this.
;;;if we decide to buy a theme template, it is possible to apply theme to asp website?
No not that I am aware of, or not in the same way you do with popular CMS solutions such as WordPress. If you are unable to create a design Perhaps you should look at re-fitting your content into a CMS such as WordPress?
;;;in asp.net we use master page as main template/design. can we do like that too in classic asp
The combination of your header, footer, CSS and .js
is the equivalent.
;;;which way is the fastest way to make changes for theme for this old web site
IMO you have two options:
1.. engage a front end specialist to code you the new look and feel. Insist that its responsive!!! Test it across you target browsers and then use the existing back end code.
2.. move to a CMS like WordPress. Choose a theme that fits the sites purpose and then work on re-fitting the content