If I stick to the same speed (e.g. do not mix say 9 speed and 10 speed), can I mix the brand of chain and missing link safely. For instance, can I use a Shimano missing link on a KMC chain, or should I only use KMC links on KMC chains and SRAM links on SRAM chains?
Shimano recommends their 9 &10 speed chains be reconnected utilizing the specific speed link pin, which is inserted with a chain tool. In my experience, there has not been any issue, including noise, using KMC's "Missing Link" or SRAM's "Power Link" to join a Shimano chain. Note that SRAM's 10 speed link is termed "Power Lock" and is suggested for use "one time only" because of the tighter tolerances present in ten speed systems. In practice, these can be used multiple times, no problem, but I base this on reading others' assessments not my own experience. Shimano's 11 & 12 speed systems use a connecting link, the "QuickLink."
Typically bike shops I frequent have KMC, SRAM, and Wipperman (whose product is called "Connex") connecting links to choose from. Prices are similar between brands within same speed categories. Connex are a bit more money it appears to me. At any rate, based on the selection, why not stay within same brand for chain and connecting link? Regarding Shimano, I use the replacement link pin but again, no problems have been noted by myself or others when a Shimano chain is joined with a connecting link from SRAM or KMC. Wipperman's design is shaped differently which may limit it to their brand of chains only. I neither have experience with nor seen anecdotal evidence to say whether a Connex link will work with other brands of chain.