data = {'IP': {'key1': 'val1', 'key2': 'val2'}}
lst = ['IP', 'key1']

current_level = data
for key in lst[:-1]:
    current_level = current_level[key]
current_level.pop(lst[-1])

Explanation

I'll use the more complex example you provided to explain how this works. The first part of the task is to get to the dictionary from which the key should actually be removed.

{
  'IP': {
    'key1': {
      'key3': {
        'key4': 'val4',
        'key5': 'val5'
        }
      },
    'key2': 'val2'
    }
}

path = ['IP', 'key1', 'key3', 'key4']

In this example, in order to remove the key 'key4', we first need to get to the dictionary that contains this key, which is the dictionary under 'key3'. If we had this specific dictionary in a variable, say d, we could just call d.pop('key4').

'key3': {
  'key4': 'val4',
   'key5': 'val5'
  }

The path to this dictionary is data['IP']['key1']['key3']. The algorithm, instead of going directly like this, starts at the root dictionary and goes one level deeper with every iteration of the for loop. So, after the first iteration current_level is data['IP']. After the next one, it becomes data['IP']['key1']. (Because since current_level is already data['IP'], current_level = current_level['key1'] is indeed the same as data['IP']['key1'].)

This process is repeated until the needed dictionary is found. That means doing this for every element in the list that specifies the path, instead of the last one, because the last one is no more a dictionary, but a key in the dictionary that we search for. (lst[:1] is Python's way of saying all elements from lst except the last one.)

Then finally, we simply pop the necessary key (the last element in the list, in other words lst[-1]) from the dictionary to which it actually belongs, the one the algorithm found in the first step.

Just change to:

dictionary = data[0]
for value in dictionary.values():
    print(value)

Here's a complete example that loads the JSON file, removes the target object, and then outputs the updated JSON object to file.

#!/usr/bin/python                                                               

# Load the JSON module and use it to load your JSON file.                       
# I'm assuming that the JSON file contains a list of objects.                   
import json
obj  = json.load(open("file.json"))

# Iterate through the objects in the JSON and pop (remove)                      
# the obj once we find it.                                                      
for i in xrange(len(obj)):
    if obj[i]["ename"] == "mark":
        obj.pop(i)
        break

# Output the updated file with pretty JSON                                      
open("updated-file.json", "w").write(
    json.dumps(obj, sort_keys=True, indent=4, separators=(',', ': '))
)

The main point is that we find the object by iterating through the objects in the loaded list, and then pop the object off the list once we find it. If you need to remove more than one object in the list, then you should store the indices of the objects you want to remove, and then remove them all at once after you've reached the end of the for loop (you don't want to modify the list while you iterate through it).

First question

However, whenever there's more than two elements and I enter anything higher than two, it doesn't delete anything. Even worse, when I enter the number one, it deletes everything but the zero index(whenever the array has more than two elements in it).

Inside delete_data() you have two lines reading i = + 1, which just assignes +1 (i.e., 1) to i. Thus, you're never increasing your index. You probably meant to write either i = i+1 or i += 1.

def delete_data():    # Deletes an element from the array
    view_data()
    new_data = []
    with open(filename, "r") as f:
        data = json.load(f)
        data_length = len(data) - 1
    print("Which index number would you like to delete?")
    delete_option = input(f"Select a number 0-{data_length}: ")
    i = 0
    for entry in data:
        if i == int(delete_option):
            i += 1  # <-- here
        else:
            new_data.append(entry)
            i += 1  # <-- and here

    with open(filename, "w") as f:
        json.dump(new_data, f, indent=4)

Second question: further improvements

Is there a better way to implement that in my Python script?

First, you can get rid of manually increasing i by using the builtin enumerate generator. Second, you could make your functions reusable by giving them parameters - where does the filename in your code example come from?

# view_data() should probably receive `filename` as a parameter
def view_data(filename: str):   # Prints JSON Array to screen
    with open(filename, "r") as f:
        data = json.load(f)
        # iterate over i and data simultaneously
        # alternatively, you could just remove i
        for i, item in enumerate(data):
            name = item["name"]
            chromebook = item["chromebook"]
            check_out = item["time&date"]
            print(f"Index Number: {i}")
            print(f"Name : {name}")
            print(f"Chromebook : {chromebook}")
            print(f"Time Of Checkout: {check_out} ")
            print("\n\n")
            # not needed anymore: i = i + 1

# view_data() should probably receive `filename` as a parameter
def delete_data(filename: str):    # Deletes an element from the array
    view_data()
    new_data = []
    with open(filename, "r") as f:
        data = json.load(f)
        data_length = len(data) - 1
    print("Which index number would you like to delete?")
    delete_option = input(f"Select a number 0-{data_length}: ")
    # iterate over i and data simultaneously
    for i, entry in enumerate(data):
        if i != int(delete_option):
            new_data.append(entry)

    with open(filename, "w") as f:
        json.dump(new_data, f, indent=4)

Furthermore, you could replace that for-loop by a list comprehension, which some may deem more "pythonic":

new_data = [entry for i, entry in enumerate(data) if i != int(delete_option)]

Assuming your data structure is stored in the variable j, you can use j['data']['ad_list'][0] to extract the first item from the ad_list key. Use a try-except block to catch a possible IndexError exception if ad_list can ever be empty.

var req = { mandrill_events: '[{"event":"inbound","ts":1426249238}]' }

console.log(Object.keys(req)[0]);

Make any Object array (req), then simply do Object.keys(req)[0] to pick the first key in the Object array.