The 6600k is OLDER than the i5-7500. You are going BACKWARDS, rather than "upgrading".
Even so, you will be likely to see a SMALL increase in performance due to the 400mhz faster boost speed, but since the all core boost speed on the 7400 is about 3.3Ghz and the all core boost speed on the 6600k is only 3.5Ghz, that 200mhz is not enough to be worth the cost of the CPU UNLESS you plan to drop a manual all core overclock on there of about 4.5Ghz which will require substantially better than stock CPU and case cooling as well as a very decent quality motherboard with a good VRM configuration.
In sum, no, that upgrade is not a worthwhile one. You would be MUCH better off to find a good used 7700 or 7700k, or simply upgrade the whole motherboard and CPU platform to a newer one and use the DDR4 you alerady have.
If you DID decide to go with the older CPU, then no, you would not need to update your BIOS.