Should you throw your crosses, jabs, uppercuts and hooks while tucking in your chin or cheek at the shoulder?

Apologies for the late answer.

While I am shadow boxing, I always wonder if it's necessary to ...

Yes. It is absolutely necessary that you do everything in your shadowboxing that you would do in your sparring sessions or fights. If you like throwing the spinning side kick when you are sparring, you must throw hundreds of them when you are shadowboxing. The idea here is simple: drill the movement, build muscle memory, and make it automatic.

Now ask yourself this: Do I need to keep my hands up and my chin tucked during sparring? If your answer is "No." then ask "Why?" in a new MA.SE question. If your answer is "Yes!" then you know "Why?" and hence you should absolutely ensure of it when you are shadowboxing.

Protect your chin at all times by tucking it in behind your shoulders (at least one), keeping your hands up (at least one) to cover your chin, or both.

I always wonder if it's necessary to protect your chin while you throw your punch combinations ...

I am going to assume you are a beginner given you state that you have not sparred with anyone until now. The answer to your question is [generally] YES! Now, you should always be conscious about protecting your chin. However, tucking in your chin behind your shoulder/s might not be suitable or appropriate at all times; doing so may adversely impact your form and power.

For example, let's say you are throwing a right uppercut (i.e., with your rear hand). When you are coming up with the right uppercut, your chin should be protected by your left arm/fist/glove. You can't tuck in your chin behind your left shoulder because it is actually pulling back (or opening up like a lever). You also can't tuck in your chin behind your right shoulder because the punch is coming from below (as opposed to the straight right/cross).

When throwing combinations, always expect the unexpected counter.

When you expect the counter, you will automatically start protecting your chin in the ways I mentioned above. You will be conscious about it, and that will help keeping you safe. You may feel that since you are throwing combinations, your opponent might not throw anything and just defend and block. But that is when you get hit by an unexpected strike.

Let's say you throw four punches (1-2-3-2); your opponent shells up and takes them. Next time you start the same combo with the Jab-Cross- ... he comes down with an over-hand right. And since you did not tuck in your chin while throwing your left hook, your third punch, you are now caught flush with that counter over-hand right and knocked out. If you tucked in your chin while throwing the left hook, that over-hand right would have grazed your head.

Tucking in your chin behind your shoulders, or keeping your hands glued to your chin is quite difficult to do when you are tired (i.e., after a few rounds of hot exchanges)!

And that is exactly why you should consciously make the effort to do it at all times during your shadowboxing. This way you train your shoulder and back muscles to be able to keep those shoulders and hands high, especially when fatigued. When you are tired, you won't have those slick head movements. Your slipping and bobbing will be much slower, and you will have to rely on the very basic defense mechanism: protect the chin with your hands and shoulders.

When certain moves don't allow you to tuck in your chin behind your shoulder/s, just consciously make the effort to protect your chin by moving your head, or by bringing your arms across your face.

It is not always possible to tuck in your chin when you are sparring/fighting. For example, let's say you throw a Jab-Cross-Left Hook-Right Body kick combo. When you throw the right body kick, your right hand should be slicing the air downwards in an angle. So you don't have your right shoulder "raised" to protect your chin as much as when you are just in your stationary fighting stance. Your left shoulder (and your left side of your whole body) is moving or turning backwards to give you the turning power for the kick. So you also don't have the left shoulder to protect your chin. What you do is bring your left arm across your face and dip your head to avoid a potential counter straight right hand.

Incorporate defense in your shadowboxing.

Often times, this whole idea goes out the window when say you're mixing up your kicks with some boxing or when you're just practicing your boxing with multiple punch combinations and head movement.

To me, it seems like when the "whole idea goes out the window" you are essentially focusing on your offense. You need to actively think about your opponent interrupting your combos and your moves.

Many beginners make the mistake of turning the shadowboxing session into a turn-based exercise. Now it's my turn to do offense {does offense for 2:30 min}; oh, well, now it's time for defense {does some parrying and blocking for 30 sec}. You need to imagine him coming at you. Mix up your offense and defense. And you will realize that you are now actively protecting your chin.

This is a better way to shadowbox: Okay, I am going to hit him with a Jab-Cross-Jab-Cross-Left Hook. Start the combo. After the first three punches your opponent moves out of the way and hits you. You raise your guard, tuck in your chin, and go back to work. Imagine your opponent interrupting your offense, and then you will automatically protect your chin, in whatever way is possible then.