One issue that plagues not just the LOTRO Mac Client but many other apps too is the new 'Gatekeeper' security feature which is included in OS X 10.7.5 & up and 10.8.x.
You can read the official information here;
Learn about Gatekeeper in OS X
Until a developer can get a unique Developer ID from Apple and use it to digitally sign their apps, then trying to install any app onto your Mac will result in an error message (check the link to see) Pretty much Apple pushing developers towards the app store etc. And a lot of legitimate apps have not yet done this for various reasons and it appears this is the case with Turbine too.
Therefore to get round gatekeeper to install the LotroLauncher.dmg you need to disable it, at least until it is installed. To do this follow the instructions posted on this site;
Temporarily disable Gatekeeper in Mac 10.8 (Mountain Lion) to allow application installations
Also, from the closed beta, it was noted as important to all the verification of the .dmg file to complete fully, as skipping it or not not allowing it to fully complete could cause issues. It did take a while to do this but important to let it fully Verify.
There may be other issues but some at least seem to have it working. Hopefully these basics may help others having issues.