Why is there no 地 in 不小心踢到椅子 ("carelessly kicked the chair")?
Strictly speaking, your translation is incorrect and that is what is tripping you up. 

不小心 has many interpretations here, and the one employed in all these examples is along the lines of "wasn't paying attention" or "was not being careful" -- this implies it is NOT actually playing the role of an adverb in the sentence. As such, 地 is not required. In essence, it is two subsequent actions. 

for example: 


  他不小心摔倒了 -- He wasn't paying attention, and fell down.


compared to


  他不小心地摔倒了 -- He carelessly fell down.


The spirit of the sentences is basically the same, but your question appears to be more about the nitty gritty details of what role 不小心 is playing in the sentence. If you wish to translate it as an adverb, then you are right, there should be a 地 there, but that's not the only way to translate 不小心. According to youdao, it can play a noun, adjective and adverb.