Resolving problems with development tools on upgrading to Mac Yosemite
I am a bioinformatician developing on the Mac and implementing on Linux. I have all of the following technologies that you mention running on Yosemite: Java, MySQL, Apache, PHP; I do not use LaTeX or Homebrew so cannot comment on them. In addition I run Tomcat, Perl CGI, Perl CGI with MySQL drivers, GD graphics for Perl, PHP with MySQL, Eclipse and Python. So in general it is definitely possible.

Your question however refers to your current installations. The answer here is that your current install of MySQL should not be affected. (If you wish you could upgrade and install the latest version available for Yosemite from Oracle's website, although this requires a free sign-up.)  Your current installation of Apache will not be available and this will also affect how PHP runs. You will also have to update your Java.

For the record, let me try to summarise what you would need to do. (I'm assuming you are starting with something as different from Yosemite as Snow Leopard. I never had Lion running - only Mountain Lion and Mavericks - so some of this may already be familiar from Lion.)

Basics


Yosemite no longer has a Developers' Tools disc. Instead you should download Xcode from the App Store. It's free, but a huge download (ca 3Gb). However you are almost bound to need it.
You need to unhide the Library folder (so you can get to the Apache install etc.) which is now hidden by default (https://support.apple.com/kb/PH18928). To make the Library folder in your home folder always visible you right-click and select View Options and then tick "Show Library Folder". However I use the main /Library folder for web development and can't remember how I unhid this. Apparently the following terminal command should work:

sudo chflags nohidden /Library/


Apache/PHP/Perl

The old option in System prefs of enabling the Apache website is no longer present for Apache 2. The following excellent article explains how to enable Apache 2 and the PHP5 and Perl modules: https://discussions.apple.com/docs/DOC-3083. Do exactly what it says. 

Java

There are three possible things you may be doing with Java, and as you don't indicate which you are interested in I shall mention them all. However the basic thing is that Apple no longer supplies its own Java, so if you need upgrades you get them from Oracle.


You write Java programs. In this case download Eclipse for Java (https://eclipse.org/) — Luna is the current version. If it launches, you have Java installed. If not, you need to download it from https://www.java.com/en/download/mac_download.jsp? . Then you are fine.
You want to run Java applications on your desktop. You will need the download above, but Yosemite may try to prevent you actually running Java applications that ran fine before. If you have used the application before or trust the author then bypass Gatekeeper by right-clicking when you launch the app and tell Apple you want to run it (https://support.apple.com/en-us/HT202491).
You want to run a Java Applet in a web browser. This is still possible but is now a real nightmare. Basically you need a signed applet and even then you may need to set your Java System Preferences specifically to allow applets from the website in question and also make sure that in Safari Prefs > Security > Allow Plug-ins is ticked and the allowed sites are listed in Website settings. Even then Safari may ask you to confirm you want to run the applet each time. The current policy on browsers now is to only allow Java applets to run if you are using the latest version of Java. I do have some self-signed applets I wrote myself that I can still manage to run. However because the whole plugin technology is being phased out, I think the Java Applet is a dead man walking. (Server-side Java is no problem though, and has advantages over PHP and the like.) More details of setting your browser up for Java applets can be found at http://humairahmed.com/blog/?p=8061 and on the Oracle/Java site.


Note about Java 8 on virgin Yosemite

If you want to run an old Java application on a brand new machine with Yosemite installed (rather than upgrading from Mavericks or earlier), and download the latest version of Java it will be (April 2015) Java 8 and the application will not run. You need to install Java 6 from https://support.apple.com/kb/DL1572. I'd download it while it's still there.