Q:

Does case matter when reading and sorting a list of items?

I want to make it easier for a user to read and be able to insert an item into a list of sorted items that can start with a capital letter or a lowercase letter.
Does the case matter in this case?
Imagine that I want to add the word bananas to the following list of fruits:
apples
Blackberries
oranges
peaches
Pears
Strawberries

or,
Blackberries
Pears
Strawberries
apples
oranges
peaches

Note that the user needs to be sure that the word bananas is not already in the list before adding it so, in the first case, the user just have to add it after apples and before Blackberries; while, in the second, the user will have to add it after apples and before oranges.
Which one will be easier for the end user?
JFTR: I can't change the capitalization of items because my use case is a convention used as a best practice for programming.

A:

Capitalization frequently matters for sorting algorithms, but seldom matters for end users. 
Trying to explain why "a" comes after "Z" to someone who doesn't know what ASCII is can make for a very frustrating user experience. 
Your first example is the best way. It makes it easier to find a specific word since there is only one alphabet instead of two. It means users don't have to know if their word was capitalized before they search for it. And it also means they can trust the system when they look for a word and don't find it instead of having so search in a second place to make sure it's really not there. 
For example, if someone was looking for "peaches" in your second example, they might stop reading at "Pears" and assume it's not in the list. And, if it's important to know that both "peaches" and "Peaches" exist in the list, they should be next to each other to make it easy to find that out. You shouldn't assume users will read the full list or even keep reading once they've "satisficed" their desire. This is even more important if the list scrolls since users may never even realize there is a second alphabet far down the list. 

A:

It matters because if it's sorted by case, you are sorting by two criteria (case and then alphabetical order) instead of 1 (just alphabetical order) thus having two implicit sublists, which adds cognitive load. 99% of the time implicit representations are terribly bad for users.
So when scanning for a lower-case word users will have to:

(first time) Search the item top to bottom (of uppercase-list)
without finding the word, then...
Realize that the first portion was for upper-case letters / locate the end of uppercase list.
(which might be confusing for some users)   
Search and find the lower case word. 

Compared to:    

Search and find the lower case word in the list.

So I'll go with the latter (the first in your example).

EDIT corresponding to the question edit:

Note that the user needs to be sure that the word bananas is not
already in the list before adding it so, in the first case, the user
  just have to add it after apples and before Blackberries; while, in
  the second, the user will have to add it after apples and before
  oranges

That shouldn't be user's responsibility, you could inline validate to prevent them to insert repeated items. Additionally you could add marks to tell him/her where the item is going to be inserted in the list.

download bmml source – Wireframes created with Balsamiq Mockups
As a side note, if the capitalization has meaning for the system but not for the user and your application runs on a browser, you can just style the text with the CSS property text-transform (thanks to @jayla for suggesting it instead of JS code), for example:
li {text-transform: capitalize}  

or
li {text-transform: lowercase}

A:

Does case matter? Perhaps.
Are the users entering the data or is this data being supplied by the system? You know your users. What will they think about the lack of consistency? If they entered the data they may not think anything about it. If they are readers of system provided information then I will be very surprised if the overwhelming majority wouldn't have a negative perception of the application and the company in general. 
After all wat wuld u think about a sight w mispellngs and texting shorthand? It may be appropriate in some circumstances but in general it's not a good idea. 
EDIT:
I would highly recommend that you pick one format or the other and stick to it. It's easy to keep the data in one format and present it to the user in another. In general, if it's a US based clientele, I would recommend that you capitalize the first letter of each item in the list. 

