I will answer this question under the assumption that you are planning to keep your custom domain name and all your article URLs intact - that is, the only change to your site that you will make is which service it is hosted by under-the-hood.
- What's the expectation after moving the site to a paid host? Will it increase the number of visitors and profit? Is it true?
There is no reason for your number of visitors to automatically increase, unless the new hosting is so much faster that less people drop off due to slow page loads. However, switching from WordPress' built-in WordAds monetization to something more flexible like Google AdWords may increase your ad revenue. This depends on what ad service WordPress WordAds is using under-the-hood (maybe someone who knows this can chime in), and what WordPress' share of the revenue is.
- Will the moving process effect on the search appearance? especially I am using a paid domain and I will move the site to the same domain name.
If done properly, the moving process will have no effect whatsoever on your search appearance. The big tricks to make it go smoothly are to (1) lower your DNS TTL for your
CNAME records well before you switch them to your new host, and (2) keep the old host up and running until the DNS has completely switched over.
- Is the problem really kind of paid hosting or something else like theme type ...etc?
Your advisors are probably recommending that you switch because a self-hosted WordPress install offers you much more flexibility than an install hosted at WordPress.com. Two of the biggest benefits are that you can use custom plugins and you can choose your own advertising platform.
- If I continue with the free plan and paid domain and just change the theme to a paid theme it will improve the number of site visitors and profit Is it enough?
You will need to know exactly why you want to move to self-hosted WordPress in the first place. Is it for the potential of increased ad revenue? Is it to take advantage of custom plugins? It's mostly about being able to do things that WordPress.com hosting doesn't allow you to do. Only you can answer this.
- I know I can perform export and import process without deleting the old articles but in this case, What is the fate of these articles in the old one? it will effect on the new one? Is google will accept it or it will consider it as duplicated?!
If you switch your domain over properly everything will go perfectly smoothly with Google. Under the hood, all that happens is that your domain name resolves to your new web host instead of Wordpress.com hosting. Google will get your new site when they visit your domain name, just as a visitor would.