Does the freshness of the ingredient impact the longevity of the cooked dish

Butter, like most fats, is actually quite resilient to microbes. The problem you are more likely to experience is rancidity. Fat goes rancid by oxidation. Exposure to light, heat, and air cause oxidation and accelerate the process of butter going rancid.

Putting butter in your refrigerator addresses the heat and light, but does nothing for the air. The paper that butter sticks are wrapped in are quite porous. You can see evidence of this as a rind that develops on your butter after a couple of weeks. Butter can also absorb odors in your refrigerator through this wrapper.

A butter crock, as shown in the accepted answer to this question actually serves to protect the butter from air, light, and heat. Butter does not need to be kept below 40 F to stay fresh. The butter crock referenced there will keep butter fresh up to temperatures of 80 F, and likely higher, but the butter will be too soft and slip out at higher temperatures.

A plastic or ceramic butter dish are similar in function to the paper that wraps butter. Ceramic is unlikely to seal at all, so the butter will still be exposed to air. Plastic, while capable of a great seal, will still trap oxygen in the container with the butter, allowing oxidation to occur.

A butter crock really does seem to be the best of all worlds. You get easy access to soft butter, with little risk. Mold will eventually grow on the butter, but this should take weeks at room temperature. If you don't use a stick of butter within two weeks, you're doing it wrong. :-D

As covered on wikipedia, raw garlic has a number of sulfur-containing compounds, among them allicin, which are responsible for the burning feeling raw garlic imparts to a dish. When cooked into food, these flavors mellow and the compounds are boiled off or denatured. Other enzymatic and non-enzymatic transformative and browning processes also take place as garlic is cooked.

Roasting garlic produces these changes in garlic before-hand, and in a more controlled way. As the garlic heats up and starts browning, the raw garlic character is lost, while sweetness from melanoidin (a product of the reaction of amino acids with sugar, per the Maillard reaction) and savoryness (from the denaturing, concentrating amino acid content) are gained. The extreme example of this is Black Garlic (discussed at some length on my blog), in which the sweet and savory content of the garlic has been processed and concentrated over the course of a month, with the flavor resembling non-sour Balsamic vinegar.

With that said, the inclusion of raw or pre-processed garlic will depend on the proportions of the two above-noted sets of flavors that are desired in a dish. If added raw to uncooked food, it will add very little savory or sweet, and plenty of aroma and burn. If added roasted to that same food, it will add mostly savory and sweet, with very little of the burn. Any further cooking that takes place after garlic's addition to a recipe will further discourage the raw flavors and aromas.