Hi I read the following sentences and I have few questions can any one hel?

Thanks in advance.

Ağızını beğenirim- why addin the (ı at the end of ağızın? 
                        why the verb time is not ın the present time?           

senin ağizın beğeniyorum - is this give the same meaning?

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boğaz ağrım var is is the same like boğazım ağrıyor? which form is most common?

thanks

Question 1, the verb beğenmek requires an accusative suffix [ı/i/u/ü] added to the verb´s object to identify what is enjoyed/adored. When you use a dictionary like the one on this site, you will see some verbs written like below

The /ı/ below the verb means that this verb needs to use the accusative suffix on the object of the verb.

Question 2, the verb is in simple present (aorist) tense, either to show this is something that happens frequently, or to indicate that this will happen.

Question 3, ağızını has 2 possible meanings here, your mouth or his/her/its mouth. By using senin ağızın there is no ambiguity, it can only mean your mouth. But, as a sentence I think you should write it as

Senin ağzını beğenirim = I (will) like your mouth

Question 4, I have a sore throat OR My throat is aching. There is a small difference, but in the end you are describing the same problem.

I can´t say which expression is more common. 

thanks a lot

"Boğazım ağrıyor." is more common. And sounds natural.

thanks for your help