Do thermodynamically favourable chemical reactions always proceed?
Because a reaction is thermodynamically favoured does not mean that it will be a fast reaction. So 'yes' they always proceed but possibly infinitesimally slowly. Temperature is the most common way of quickening a reaction. Of course a catalyst also does so but this generally by changing the way the reaction occurs, i.e. the mechanism is different using a catalyst but the product is the same. Enzymes are natures catalysts, in synthetic chemistry (and in you car exhaust) metal based compounds are often used.

All reactions have an activation energy $E_a$ between reactants and products and a small increase in the size of this can slow a reaction exponentially. Experimentally, the rate constant is generally found be of the form $k=k_0 exp(-E_a/RT) $, which is the Arrhenius equation, with R the gas constant and T the temperature in Kelvin. 

The Boltzmann thermal distribution explains why this type of equation applies. This shows that as temperature increases molecules have more energy and so surmount the activation energy more frequently and so reaction rate increases,