How can it be converted to a pure mainland Chinese version?
Using Google Chrome, there is an extension for translating websites. In the past this used to be extremely popular, until English only speakers realized the translations were not only horrible, they were machine translations. For your case, you can use machine translation if you have either Simplified or Traditional Chinese because in essence they are the same, only the writing is different.
I mean are the differences just in typing?
The differences are just in typing. However, a few exceptions to this rule applies when you are talking about the different dialects of Chinese (Standard Madarin is the one all Chinese speakers know, Cantonese is slightly different, Taiwanese is also different).
Are there any difference in the vocabulary?
Vocabulary is basically the same. Slang wise, they will be different. There might be some slight variations in how common certain words are used, but overall, Mainlanders can communicate with those in Taiwan, Hong Kong, and even communicate among chinese speakers in Indonesia, Malaysia, Singapore.
Is there any service on the net which gets a traditional word (or a word typed in traditional Chinese) and then gives you its simplified version?
Google Translate (the plug in) will quickly help translate from Simplified to Tradition. Granted, you have either one or the other as the source language.
Do I have to apply grammatical changes, too?
Grammar is the same. The only problem occurs when users use local slang.
Do simplified Chinese (People who are living a mainland China) understand traditional Chinese easily? What about vise versa?
Traditional Chinese is used only in Taiwan now. That being said, Taiwanese can read simplified, (maybe not as fast or fluent as reading traditional). You can use this Chinese SE as an example. There are many users on this site that use simplified/traditional/both.
Which one is used wider in the world (traditional or Chinese)? (I mean for promotional purposes is it better to translate a passage into simplified or traditional to be understandable by more people?)
In terms of promotion, it would be simplified. However, this is talking about marketing. I heard you want to promote on facebook. Is facebook blocked in China? Yes it is blocked, but it is so easy to get around to the point, no, it's not blocked to anyone who has a college degree. Is this a good marketing strategy? I won't comment on this because this starts to become irrelevant to Chinese SE.