It is a stretch, but here goes:
Do not worry, seeing me upside down is still me.
Upside down, it reads 60109 also.
Rearrange me, the new number is zero. (credits: BlueFire)
Rearranging 60109 could make it so that it becomes 60 - 60, the minus being the 1 and the other 60 being the upside-down 90. Result is 0.
Take 3 digits from me then rearrange them, you will be very very sick
Take '6' reverse it to get '9', take 0 and 1 put them next to each other to get another '9'. Result is '999' the emergency number.
Take 3 digits from me then rearrange them, it is the beast
Similar to the previous one, take '9' reverse it to get '6', put '1' and '0' together to get '6' and you get '666' the devils number.
Take 4 digits from me then rearrange them, add the rearranged number with the unused digit, the result is 10000.
Not sure about this one, could be using imagery (separating the '6' into '1' and '0', and the same for '9' gives 0000 and "add" the '1' to get "10000")
My middle is a sparator of 2 same numbers, lets call it jack.
Jack could be '010'
Do not worry, seeing jack upside down is still jack.
010 upside down is 010
Jack is 2 digit number, glue it, it is a bicycle part.
010, which is 2 digits ('10') added together could resemble a bicycle wheel and chain.
I know its far-fetched, but its a theory.