#You need a single hand free for both Material and Somatic Components to be met The Rules on Spellcasting specify how Material and Somatic components work:
If a spell requires a somatic component, the caster must have free use of at least one hand to perform these gestures.
A character can use a component pouch or a spellcasting focus (found in “Equipment”) in place of the components specified for a spell....A spellcaster must have a hand free to access a spell's material components -- or to hold a spellcasting focus -- but it can be the same hand that he or she uses to perform somatic components.
##Specific Questions Using these guidelines, let's look at the questions presented:
Question 1
If a focus is merely worn, can a wizard still cast his spells?
No - as stated under Material Components, a spellcaster must have a hand free to access...to hold a spellcasting focus.... In order to cast with a Material Component, the focus can't simply be worn, it must be held.
To compare, classes that can use Holy Symbols (paladins and clerics) have the clear path to displaying their Holy Symbol as a Material Component on a worn item such as a shield (PHB, 151). This case gives a very clear example of needing to be holding the Focus.
To use the symbol in this way, the caster must hold it in hand, wear it visibly, or bear it on a shield.
Question 2
If it fits, can the wizard hold a component or focus in the same hand that performs somatic components?
Yes - and fitting really isn't a necessary requirement. The rule quoted above regarding Material Components explicitly states that it can be the same hand that he or she uses to perform the somatic components.