Sentence 2 is correct. Based on how you stated it, Sentence 1 is also correct.
Sentence 2 - "Kailangang kumain si Ana" = "Si Ana ay kailangang kumain" = "Kailangang si Ana ay kumain" = Ana need/has to eat. The 3 sentences are all grammatically correct. If we are going to move the "na" from after "kailangan na/kailangang" to after "Ana" - Kailangan SI Ana na/Anang kumain - it should instead be "Kailangan NI AnaNG kumain", which is what Sentence 1 is.
Sentence 1, however, is only correct if it is stated the way you did, i.e., the sentence starts with the verb. The "na" linker is needed in "kailangan na/kailangang kumain" because it functions like the "to" that links the verbs "need" and "eat" to form the phrase "need to eat".
The main verb of the sentence is "kailangan". Usually, the actor of the verb should follow it immediately. Hence, we would more often state that the sentence as "Kailangan ni AnaNG/Ana NA kumain - the "na" attaches to Ana instead. (This is what I meant at the end of my comment re Sentence 2 above).
The problem with the sentence "Kailangan ni Ana NA kumain" will appear if we are going to state it in the subject-predicate form where we need the "ay" as their partition. What appears on either side of NA may serve as the subject or the predicate.
Ang kailangan ni Ana AY kumain (What Ana needs is to eat).
Ang kumain AY ang kailangan ni Ana (Eating is what Ana needs). - "Kumain (to eat)" becomes the gerund "eating".
Notice that in both sentences, the "na" is no longer used because the phrase "kailangan na kumain" is no longer there. That is why I mentioned above that your Sentence 1 is only "correct" in that form, i.e., when the two verbs have to go together and be linked with a "na".
I am not clear about what you are referring to by "usage of kailangan (NG or ANG?)" If what you meant was about the change of "ng" to "ang" when the actor becomes the subject of the sentence, then that would only apply to Sentence 2. If we'd replace "si Ana" with a common noun, "Kailangang kumain si Ana" becomes, for example,:
Kailangang kumain NG babae. = ANG babae ay kailangang kumain.
You also mentioned that you found fewer cases of "kailangan siyang". That's because when "siya" is the actor of two linked verbs, we would normally use "niya"; or we use "siya" when "siya" is the object of the verb.
In the first four citations under "kailangan siyang":
1. "... kailangan siyang manood ng ads..." - I believe that most of us would say this as "kailangan NIYANG manood ng ads... (he/she needs to watch ads)"
2. "... or may kailangan siyang bitbitin... (or there is something that he must carry with him)" - The "may" before "kailangan" requires that "siya" is used. We don't say "may kailangan kong/mong/niyang/nating/ninyong/etc. + verb". What we say is "may kailangan akong/kang/SIYANG/tayong/kayong/etc. + verb".
3. "... Kailangan siyang mag-trabaho, anak." - Again, I think that most of us would use "niyang" instead of "siyang" here. Also, in the absence of context, to use "siyang" may suggest that someone needs to work instead of doing nothing; while using "niyang" is normally understood as the need to earn a living as the head of the family.
4. "... Kailangan siyang mailigtas sa Hawaii... (He/She needs to be rescued in Hawaii)" - "Siya" is the object of the verb, the actor would be the rescuer.
There are more of the "kailangan niyang" because, as I mentioned above, the actor of the verb "kailangan" usually follows it immediately, i.e., the actor comes before the second verb.
If you search only for "kailangang", you will see that in the majority of the citations, the second verb comes right after it. And if you'd place the actor pronoun between the 2 verbs, they'd be "kailangan kong/mong/NIYANG/naming/etc." followed by the second verb. That's why "kailangan niyang" is more common.
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