What are the requirements to run LiveSearch?

At least your webserver shpuld be able to handle PHP-files, so you need PHP 5.x or higher.
Your webserver is acting like a client browsing the output of your website, so your webserver should be able (and the allowance) to access your website with either enabled allow_url_fopen.
Optional to create smaller thumbnail images if needed you'll need an installed GD-Library and if you like to search through PDF documents you'll need pdftotext on a Linux machine. There are ways to get pdftotext running on Windows Servers too, download and information can be found on http://www.foolabs.com/xpdf/download.html .

I have a static website, can I use LiveSearch?

Even if you just have HTML pages you would be able to use LiveSearch as long as you meet the requirements that you are abe to run PHP scripts and your webserver is able to access your website. You only need to built a search results page, which should be a PHP-File.

Why it took so long time to display the search results after starting each search?

Everyting is depending on the settings in the class. LiveSearch operates in two steps, indexing/caching files and searching in the cached files. The searching process itself should be fast. Caching time is depending on the scope of your website and would take some time.
But how to increase the speed?
Just adapt the $cachetime to an applicable value or disable it if there are no changes or you like to run the caching process not during performing a search i.e. with the LSManager.

How to get rid of incorrect displayed special charcters?

You could try to act/deact $utf8DecodeResults and/or use the ISO8859-15 class file. And don't forget to recache after changes are made.

How to recache/reindex after I made some changes to my website?

If you have set the $cachetime to an value higher 0 the indexing/caching process will be automatically started when the $cachetime is expired (-1 will index every time and i.e. 2 would cache with the first search 2 hours after the last indexing process).
But if you disabled $cachetime by setting it to 0 or are using a really high value you could either use the LiveSearch Manager when you like to reindex/recache the website or (just one example) you could do it periodically if you are able to let a cronjob call a website.
// myPeriodicIndexer.php
  $LiveSearch = new LiveSearch();
  return true;
and on the webserver you could install a cronjob calling
wget --spider -T0 http://yourdomain/myPeriodicIndexer.php
something like
# website indexer (daily at 2 am)
* 2 * * *  wget --spider -T0 http://yourdomain/myPeriodicIndexer.php
or you can use an online cron service like listed on http://www.cronjobservices.com/ .