Can a neutral tone appear on a non-final syllable?
Taking the definition of "word" as "an entry in CC-CEDICT", the answer to the first question is Yes.

I have a little script (I could share it if there's any interest) that I adapt to go through CC-CEDICT, looking for words that match a few criteria. Here's the output I got for 2-character words starting with the neutral tone:


  nb matches: 8
  
  nb tone patterns: 5
  
  4   5, 5
  
  1   5, 1
  
  1   5, 2
  
  1   5, 3
  
  1   5, 4
  
  5, 1   嗯哼 嗯哼 [en5 heng1] /uh-huh/
  
  5, 2   個頭 个头 [ge5 tou2] /(coll.) (suffix) my ass!/yeah, right!/
  
  5, 3   得很 得很 [de5 hen3] /very (much, good etc)/
  
  5, 4   的話 的话 [de5 hua4] /if (coming after a conditional clause)/
  
  5, 5   出來 出来 [chu5 lai5] /(after a verb, indicates coming out,
  completion of an action, or ability to discern or detect)/
  
  5, 5   著呢 着呢 [zhe5 ne5] /comes at the end of the sentence to indicate
  a strong degree/quite/rather/
  
  5, 5   起來 起来 [qi5 lai5] /(after a verb) indicating the beginning and
  continuation of an action or a state/indicating an upward movement
  (e.g. after 站[zhan4])/bringing things together (e.g. after 收拾[shou1
  shi5])/(after a perception verb, e.g. 看[kan4]) expressing preliminary
  judgment/also pr. [qi3 lai5]/
  
  5, 5   麼麼 么么 [me5 me5] /kissing sound (slang onom.)/


Note: the output starts with some statistics (number of matches, number of tone patterns) along with the different tone patterns (aka tone contours) ordered by frequency. 

There are too many 3-character words containing a neutral tone not at the end for me to put them here - there are 445 of them - but here are the first lines of the output of my search.


  nb matches: 445
  
  nb tone patterns: 22
  
  49   4, 5, 4
  
  48   1, 5, 4
  
  29   2, 5, 4
  
  27   2, 5, 2
  
  26   3, 5, 4
  
  26   4, 5, 3
  
  25   1, 5, 1
  
  23   2, 5, 1
  
  21   1, 5, 3
  
  21   3, 5, 3
  
  18   1, 5, 2
  
  18   3, 5, 1
  
  18   3, 5, 2
  
  17   4, 5, 2
  
  16   4, 5, 1
  
  14   4, 5, 5
  
  13   1, 5, 5
  
  13   3, 5, 5
  
  11   2, 5, 5
  
  10   2, 5, 3
  
  1   5, 3, 3
  
  1   5, 4, 5
  
  1, 5, 1   吃不消 吃不消 [chi1 bu5 xiao1] /to be unable to tolerate or
  endure/to find sth difficult to manage/
  
  1, 5, 1   吃不開 吃不开 [chi1 bu5 kai1] /be unpopular/won't work/
  
  1, 5, 1   吃得消 吃得消 [chi1 de5 xiao1] /to be able to endure (exertion,
  fatigue etc)/to be able to afford/
  
  1, 5, 1   吃得開 吃得开 [chi1 de5 kai1] /to be popular/to be getting on
  well/much in demand/
  
  1, 5, 1   夾肢窩 夹肢窝 [ga1 zhi5 wo1] /armpit/also written 胳肢窩|胳肢窝[ga1 zhi5
  wo1]/
  
  1, 5, 1   媽媽桑 妈妈桑 [ma1 ma5 sang1] /mama-san, middle-aged woman who
  runs a brothel, bar etc (loanword from Japanese)/madam/
  
  1, 5, 1   尖子生 尖子生 [jian1 zi5 sheng1] /top student/
  
  1, 5, 1   書記官 书记官 [shu1 ji5 guan1] /clerk of a law court/
  
  1, 5, 1   梔子花 栀子花 [zhi1 zi5 hua1] /cape jasmine (Gardenia
  jasminoides)/
  
  1, 5, 1   椰子汁 椰子汁 [ye1 zi5 zhi1] /coconut water/
  
  1, 5, 1   椰子貓 椰子猫 [ye1 zi5 mao1] /Asian palm civet (Paradoxurus
  hermaphroditus), also called toddy cat/
  
  1, 5, 1   歐巴桑 欧巴桑 [ou1 ba5 sang1] /older female/woman of mature years
  (Japanese loanword)/
  
  1, 5, 1   滴答聲 滴答声 [di1 da5 sheng1] /tick (tock)/
  
  1, 5, 1   玻璃心 玻璃心 [bo1 li5 xin1] /(slang) overly sensitive/butthurt/
  
  1, 5, 1   玻璃杯 玻璃杯 [bo1 li5 bei1] /drinking glass/
  
  1, 5, 1   玻璃砂 玻璃砂 [bo1 li5 sha1] /siliceous sand (geology)/
  
  1, 5, 1   玻璃鋼 玻璃钢 [bo1 li5 gang1] /glass-reinforced
  plastic/fiberglass/
  
  1, 5, 1   生意經 生意经 [sheng1 yi5 jing1] /knack of doing business/business
  sense/
  
  1, 5, 1   眵目糊 眵目糊 [chi1 mu5 hu1] /(dialect) gum (in one's eyes)/
  
  1, 5, 1   胳肢窩 胳肢窝 [ga1 zhi5 wo1] /armpit/also pr. [ge1 zhi5 wo1]/also
  written 夾肢窩|夹肢窝[ga1 zhi5 wo1]/
  
  1, 5, 1   芝麻包 芝麻包 [zhi1 ma5 bao1] /sesame bun/


As for the second question about the neutral tone being lexical or sandhi, I think the answer is "lexical". I'm not aware of any sandhi rule governing the neutral tone.