The license provisions of the LGPL only apply if you are modifying the code and then distributing it.
#1 isn't modifying anything, so you have no problems there.
#2 isn't distributing the modified code, so you are OK there as well. Take note that (for example) modifying LGPL-ed Javascript code and using it on a web page counts as "distribution" since it leaves your server and is transmitted to the user. PHP code is usually not counted as "distribution" since it is processed/transformed before it is transmitted and the page the user receives is fundamentally different than the LGPL-ed PHP source.
#3 will require you to make your modifications available. You don't have to distribute the sources with the client software, having the modified source available on your website and including a note with the client software indicating where they can download the source is usually sufficient. The main point is that they can get to it if they want it. This shouldn't affect the license of your client software, though. As long as you link to the library the same way before and after your changes, your client is unaffected by your changes to the library source.
#4 is essentially the same as #3. Taking part of the code and forking it off into a new project falls under the umbrella of "modifying the code". In this instance you would probably keep the same LGPL license, but you have the option of changing to another license type as long as it is compatible (usually this means the new license cannot be more restrictive than the old one).