How can I seal the large gap around my fireplace in a way that is safe and prevents air inflows?
You can buy concrete in a caulk tube. I actually just finished doing this same thing to my fireplace that I remodeled, here's the product I used for the smaller cracks that I wasn't able to do with my bulk batch of concrete:

http://www.homedepot.com/h_d1/N-5yc1v/R-100357212/h_d2/ProductDisplay?langId=-1&amp;storeId=10051&amp;catalogId=10053

After seeing the photo of the gap in question a high heat silicon sealant would give a better matching finish:

http://www.amazon.com/Meeco-High-Temperature-Silicone-Sealant/dp/B002CTRWRQ/ref=pd_sim_dbs_indust_2