How many words can a pen with half refill write?

I welcome the day with a show of light, I stealthily came here in the night. I bathe the earthy stuff at dawn, But by noon, alas! I'm gone.

You are given four results as below:
1111 = F
2222 = E
3333 = T
Then, can you find out how to code 4444?

Three people check into a hotel. They pay $60 for the rent of the room. After they check-in, the manager realize that the rent for the room is $55. So, he gives $5 to the bellboy and asks him to give it to them. The bellboy thinks that it will be difficult for the three people to share $5 among them and seeking the personal benefit, he pockets $2 and gives the remaining $3 to them.

Now, each person paid $20 and got back $1. In this manner, each of them paid just $19 which totals to the amount of $57. The bellboy has $2 with him and adding them, we get $59. So where is the remaining $1?

You can find some missing letters in the picture. By placing two particular letters in the spaces, you can form a nine lettered word beginning from one of the corners and going clockwise direction to the middle. Can you find out the letters and the word?

Nodes are shown below and you need to connect them based on the following rule: Every node can be connected to the number of nodes inside them, i.e. the first node there is a value of 1 which indicates that the first node can connect to exactly one node only.Can you do it?

If,

20 % 5 = 24

21 % 7 = 33

42 % 6 = 67

Then;

66 % 6 = ?

What is harder to catch the faster you run?

Be active and find the killer in the picture Riddle below.

A finger goes in me. You fiddle with me when you're bored. The best man always has me first. What am I?