What are the primary sources that discuss the rule of taking a meat spoon and using it in my dairy coffee. Would it matter how long ago the spoon was used?
Assuming the coffee wasn't heated in that mug, then it has the status of Kli Sheni, which the Shulchan Arukh rules (105:2) does not transfer flavor bedieved (though lechatchila it should be avoided). However, see the gloss there of Rabbi Akiva Eiger who posits that since metal utensils can be Kashered, it is not considered a bedieved situation since you can just Kasher them. Ceramic utensils however (such as, likely, the mug) would be fine since there is no further recourse. Accordingly, one should not put the meat spoon in dairy coffee, but if one did then coffee is fine to drink and the spoon might need to be Kashered before further use.
If the coffee was heated in that mug and is still hot, then we're back to the regular rules of YD 94: if the spoon wasn't used for hot meat in the last 24 hours then the coffee and coffee mug are kosher, and if the spoon was used for hot meat in the last 24 hours then the coffee and coffee mug are only kosher if the volume of the coffee is 60-times (some say it's [sixty times the number of times the spoon was dipped in]-times) the volume of the part of the spoon dipped in the coffee. Either way the spoon must be kashered.
(These are some primary sources. CYLOR with the details of any actual case as there can be many ways things might differ from the positions and cases mentioned here.)