verbs - Usage of lie and lay

I'm preparing for some competitive exam and I'm reading an English grammar book, in that I encountered a test on lie and lay, but I do have a doubt on two questions, I don't know whether I'm right or not but let me tell you about my knowledge on usage of lie and lay

1. Lie is always used as intransitive verb in all it's forms:
   
   lie lay lain lying


2. Lay is used as a transitive verb and also used in passive voice:
   lay lay laid laying

Now the two questions and my answers, but I do not know whether my answers are correct or not:

1. (Laid,Lain) on its side the child was able to breathe better.

My answer is "lain" because I think there's a need of intransitive verb so obviously it's "lain".

2. (Laying,Lying) on its side, the child was able to breathe better.

My answer "lying".

Answer

If you are unsure about the right form, rephrase the sentence with a different verb - the two typical substitutions, however inelegant, could be "place, position" vs. "recline, rest".

For your sample sentence (regarding the given choice of verb form) we get

1. 
   

   With the past participle, indicating passive¹:

   
   

   
   

   (Having been) Positioned on its side....

   
   

   
   


2. 
   

   Using the continuous form:

   
   

   
   
   

   Resting on its side...

   
   

   
   

Now picking the matching form of lie / lay from your handy list, you get:

1. 
   

   
   

   Laid on its side...

   
   

   
   
   

and

2. 
   

   
   

   Lying on its side...

   
   

   
   

And yes, you can also say that sentence 2 has no object, hence must be a form of "lie".

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¹ We are looking at a participle clause with the past participle here, which have a passive meaning