To answer All your Questions:
What happens if you don't have any professions?
You will receive Professions that are useful to your character class. The following Professions are automatically Learned depending on your Armor Type:
- Cloth Armor: Tailoring and Enchanting
- Leather/Mail Armor: Skinning and Leatherworking
- Plate armor: Mining and Blacksmithing
As you can see, those Professions are decent, however most of the time it's better to log in and learn 2 new Professions of your choice.
How lang can you save the Level Boost
Once you've paid for it you can basically wait forever to use it. Currently there is no time limit however this might change in the future (although it's rather unlikely to happen).
How do I apply the Level Boost?
You will see an Icon in the Top right corner of your Character Select screen. If you click it you are prompted to choose the character. After that you have to select the Talent specialization of your char (or fraction if you are a Pandaren). If you've done all this you are asked one more time to confirm your choice and if you click yes your level 90 Character is ready.
Sources: Character Boost FAQ and personal experience