Can any solid of low heat capacity release or gain energy slower than another solid of higher heat capacity? If our sense of contact temperature is influenced by the thermal conductivity of the material: 
Can any solid material with a low heat capacity exist that feels closer to human body temperature than another solid material with a higher heat capacity; where both materials were previously kept in either a mundane oven or freezer for a sustained period?

I start thinking about whether heat capacity will always be symmetrical for gaining and losing energy and then I realise I've moved beyond my level of Physics knowledge.
 A: 
Can any solid material with a low heat capacity exist that feels closer to human body temperature than another solid material with a higher heat capacity; where both materials were previously kept in either a mundane oven or freezer for a sustained period?

Let me rephrase to:

Is there any solid which disobeys the inverse proportionality of thermal conductivity and specific heat capacity?

Consider $1000kg$ of wood and $1000kg$ of aluminium, both at $320K$ (very warm). At the instant you place a finger on such large thermal masses, your perception of temperature comparison is dependent on heat conductivity of the materials, not their heat capacity (their masses are so large compared to your finger, their temperature is almost constant depsite losing heat to your finger). Using such large masses and (equal masses for that matter) is necessary since otherwise I can instantly answer yes to your question by giving you 100g of wood and 1g of gold (beaten to the same surface area of the wood) just taken from the freezer and you would perceive gold being closer to body temperature than the wood after a second. So lets define the question by specific heat capacity, and instantaneous perception of heat transfer.
To answer it though, there is in fact no metal which disobeys this relation due to the electron sea being the majority carrier of kinetic energy in the bulk metal. Their having large mean free paths and low masses allow them to attain very high velocities (which is a property of high temperature) and therefore are able to transfer energy quickly in the bulk material. In other words, if metals used anything heavier to transmit heat, like their nuclei, it would not only take much more heat to accelerate them to the same velocities the electrons could attain (resulting in higher heat capacity), but the rate at which that kinetic energy is transmitted across the material is accordingly slower (lower thermal conductivity). In fact the lattice of metal nuclei do in fact contribute to both properties via phonons not translational kinetic energy like in gases, but phonons are still greatly superseded by the effect from electrons. Therefore the inverse relation between thermal conductivity and heat capacity is valid for metals.
What you are looking for is a non conductor with both higher heat capacity and thermal conductivity than a conductor. For that I give you diamond (figuratively...I can't afford one), which has a specific heat capacity of $0.5 J/gK$, higher than that of any metal denser than vanadium (which is almost all of them), but has a thermal conductivity of $>900W/mK$, trumping silver's $421W/mK$ which is tops for all pure metals. 
Indeed, $1kg$ of silver would feel much closer to body temperature than $1kg$ of diamond (that's alot of diamond!) despite diamond having a higher heat capacity.
A: Yes. Objects from the freezer that feel unusually cold are good at removing heat from your body (and conversely, they will feel hot if they are put in an oven). When you touch a cold object, the part right near your skin (the "surface" layers) warms up and your skin cools down. It's the cooling of skin that you feel.
If the object has a high heat capacity, it will feel colder because it will take more heat to warm up the surface. If it has more heat conductivity, it also feels colder because the interior can more easily pull heat away form the surface, keeping the surface cool.
Wood has more heat capacity but far less thermal conductivity than aluminum or diamond. Wood's lesser conductivity overwhelms it's greater capacity so wood feels less cold/hot. The exact rate of heat transfer is a complex issue (see the heat equation), however.
