How to quickly copy the current editing file name or full file path in Eclipse (Luna)?

You can use start explorer (alternate link) or Copy path plugin.
Short cut key to copy path to clipboard in star explorer is : Ctrl+Alt+ C

Standard eclipse eclipse doest have such key. You can select resource in Package/Project/Navigator view and press Alt+Enter to open property dialog and then copy path from here.

I am using Eclipse Java EE IDE for Web Developers.Version: Luna Release (4.4.0) Build id: 20140612-0600

You can right mouse click on the file and select "Copy Qualified Name".

For this particular example, the path in clipboard is

/TestingProject/src/com/website/testing/App1.java

If using Eclipse 4.14 or later, from Dec 2019 or later, neither StartExplorer nor Path Tools nor Copy as Path works.

Instead, use EasyShell, which works great!

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Additional note: as far as good Ecliplse plugins go in general, I also highly recommend the DevStyle plugin, set to "Dark Gray (Darkest Dark)" Workbench theme, with the DevStyle "Editor theme" (syntax highlighting) set to Sublime Text 3 (Monokai), by Jeremy Shepherd <-- Update Feb 2020: dead links. Instead, I've copied and pasted the contents of Jeremy Shepherd's .xml file at the bottom of this answer.

Sublime Text 3 (Monokai) - by Jeremy Shepherd--theme-25808.xml:

<?xml version="1.0" encoding="utf-8"?>
<colorTheme id="25808" name="Sublime Text 3 (Monokai)" modified="2014-04-11 19:52:55" author="Jeremy Shepherd">
    <searchResultIndication color="#757575" />
    <filteredSearchResultIndication color="#757575" />
    <occurrenceIndication color="#000000" />
    <writeOccurrenceIndication color="#000000" />
    <findScope color="#111111" />
    <deletionIndication color="#D25252" />
    <sourceHoverBackground color="#000000" />
    <singleLineComment color="#75715E" italic="false" />
    <multiLineComment color="#75715E" italic="false" />
    <commentTaskTag color="#CCDF32" italic="false" underline="true" strikethrough="false" />
    <javadoc color="#76725E" italic="false" />
    <javadocLink color="#76725E" italic="false" underline="true" strikethrough="false" />
    <javadocTag color="#FD2971" italic="false" />
    <javadocKeyword color="#C2BFA6" italic="false" />
    <class color="#56D8F0" bold="false" underline="false" strikethrough="false" />
    <interface color="#D197D9" />
    <method color="#FFFFFF" />
    <methodDeclaration color="#7BE12A" />
    <bracket color="#D8D8D8" />
    <number color="#7FB347" />
    <string color="#E7DD6C" />
    <operator color="#D8D8D8" />
    <keyword color="#F12971" bold="false" />
    <annotation color="#FFFFFF" />
    <staticMethod color="#BED6FF" />
    <localVariable color="#E7F8F2" />
    <localVariableDeclaration color="#F12971" />
    <field color="#7BE12A" />
    <staticField color="#EFC090" />
    <staticFinalField color="#EFC090" />
    <deprecatedMember color="#D25252" underline="false" strikethrough="true" />
    <enum color="#7BE12A" />
    <inheritedMethod color="#BED6FF" />
    <abstractMethod color="#BED6FF" />
    <parameterVariable color="#79ABFF" />
    <typeArgument color="#BFA4A4" />
    <typeParameter color="#BFA4A4" />
    <constant color="#EFB571" />
    <background color="#272822" />
    <currentLine color="#3E3D32" />
    <foreground color="#F8F8F2" />
    <lineNumber color="#72736A" />
    <selectionBackground color="#757575" />
    <selectionForeground color="#D0D0D0" />
</colorTheme>

Sublime Text 3 (Monokai) - by Jeremy Shepherd--theme-25808.epf:

Too long to post here. See my dotfiles project to download it instead.