NOTE: All I will talk about in this answer is when related to a verb and in an Affirmative sentence! There are other meanings in other situations.
1. Difference between 正在 and 在 (before a verb).
正在 and 在 are almost the same, just like "I am"/"I'm" and "He is"/"He's"
Q:你在做什么？What are you doing?
A:我在吃午饭。I'm eating lunch.
A:我正在吃午饭。I am eating lunch.
Both are OK.
2. 呢(after a verb) is a common particle in Chinese which is used to express feelings in the sentence.
呢 shows less respect if added in to a sentence. Usually happens in a conversation, to a same-aged friend, or someone younger.
Boss: 你可以尽快处理好这个问题吗？Can you solve this issue ASAP?
Employee: 我在吃午饭，我吃完会尽快解决它的。I'm eating lunch，I will fix it ASAP after lunch.
Employee: 你可以帮我处理这个问题吗？ Can you help me solve this issue?
Boss: 我在吃午饭呢，等会再说。I'm eating lunch, will talk about it later.
Answer to the Question:
Lieutenant: 李现在在干什么？去让他过来见我。What is Lee doing right now? Go ask him to come here to see me.
Soldier: 他正在睡觉。我现在去叫醒他。He is sleeping. I'm going to awake him up.
Lee: 中尉现在在干什么？我需要和他说话。What is Lieutenant doing right now? I need to talk to him.
Soldier: 他正在睡觉呢。你最好一小时后再来。He is sleeping. You better come an hour later.