It will help to think of spelling and pronunciation separately.
- 6 in all multi-digit numbers are written as 육. So 16 is written as 십육 and pronounced as /심뉵/.
- When the word is read as a string of single digit numerals, 6 is written 륙 only if it comes after a syllable ending with ㄹ or a vowel:
A. in strings of digits as in phone numbers. 266-1600 (이륙육(의) 일륙공공).
B. in informally or-ed numbers. 오륙 명 = five or six people (doesn't mean fifty six).
C. when numbers form a proper name. 오륙도 (an island), 오일륙 (5/16/1961), 십이륙 (10/26/1979).
(Note that 오일륙, etc. are treated like names referring to the event on that date.)
The reason for the two different spelling is because the original pronunciation is 륙 but Korean phonetics shun the ㄹ sound at the beginning of words or constituent sub-words of a compound word. So it changes to the variant 육 in most positions except when it attaches to another part like a suffix. The digits in multi-digit numbers are treated like head positions (since every digit is equally important), so numeric values only have 육.
- 육 is written and pronounced as /륙/ after a syllable ending with ㄹ (2-A above).
- 육 is pronounced as /뉵/ after a syllable ending with ㄱ,ㄴ,ㄷ,ㅁ,ㅂ,ㅇ,ㅈ, etc (but only ㄱ,ㄴ,ㅁ,ㅂ,ㅇ actually exist in numerals). e.g. 106 (as a number) = 백육 = /뱅뉵/. 106 (as individual digits) = 일공육 = /일공뉵/.
The reason for the above has to do with general Korean phonology. For example, ㄱ followed by an 이-sound becomes /ㅇ-ㄴ/ sound if it happens across a morpheme or concept boundary. So 한국 요리 is pronounced /한궁뇨리/, while 확인 is just /화긴/ and not /황닌/ because 확 and 인 don't form independent concepts. There is a whole slew of these (rather complex) phenomena that apply to all 이-based vowels (이,야,얘,여,예,요,유) and not just 육.
Is 106 pronounced 백육?
A: /뱅뉵/ if it's a single number, /일공뉵/ if it's a string of individual digits.
(Note 백육 is pronounced either /뱅뉵/ or /배귝/ and no other way. "백육" doesn't represent pronunciation)
Does the spelling 륙 still exist, as in 십륙, or is it spelled 십육, although pronounced [심뉵]?
A: 륙 doesn't exist in mathematical numbers like 16 (for this, 십육 is the only correct spelling), but it exists in numeral lists and names (오륙도, 십이륙).
Is the [뉵] pronunciation of 6 used only immediately after 십?
A: No. There are other cases like 106 (/뱅뉵/), 1,600 (/천뉵빽/), 16,000 (/만뉵천/), etc.
Lastly I'd like to mention that all this is complicated stuff (kind of an anomaly in the language) most native speakers make mistakes about (and don't even care much). And people will understand you even if you don't write or say them exactly right, so don't stress yourself too much about it.
