If I apply a theme on all songs in order to all have the same font. Will I be able to change the background of the songs so it always keep the new background or the theme will always overwrite my changes?

Once you edit an EW item (song/presentation/scripture) your changes remain with the item. So the closer the settings are made to item the higher precedence they have. I'll use a song in the following explanation.

Order of application of settings, if setting has been modified:
[list:276ckvlv]
1. Song specific settings (Font, Background, etc.)
2. Default theme settings (Font, Background, etc.)
3. Global font settings (Font)[/list:u:276ckvlv]

Example (without Default Song Theme):
[list:276ckvlv]
Global Font - Arial
Any Theme Font - Lucida Sans
Song Font - Segoe UI[/list:u:276ckvlv]
New song with no theme will use Global font (Arial)
New/Existing song with Any Theme applied will use theme font (Lucida Sans); changes to Global font nor Theme font won't affect it
New/Existing font changed to Segoe UI, changes to Global font won't affect it

It gets a little confusing with the Default Song Theme:
[list:276ckvlv]
Default Song Theme Font - Calibri[/list:u:276ckvlv]
New song will use Default Song Theme font (Calibri)
New/Existing song with Any Theme applied will use theme font (Lucida Sans); changes to Global font nor Any Theme font won't affect it
New/Existing font changed to Segoe UI, changes to Global font nor Default Song Theme font won't affect it

Note: You can also set a Default Background which would follow same precedence rules as Global Font; it is overridden by Theme and Song settings.