Complete the missing parts in the sentences below.

1) Yaşlı kad....  ev.....  besle....   kedi  d..  o..  tırma....

 The cat which the old woman feeds in her house scratched her yesterday.

2) Arka.......  araba  b...  geç...  is.....

The car behind us wants to overtake us.

3) Bahçe....   ba..  ağaçlar  yaprak......  dök....  başla..

Some trees in the garden have begun to shed their leaves.

4) Ev....  bir kedi  bes.....  yaş..  kadın ağ.......   bi..  gel..

The old woman who feeds a cat in her house came us by crying.

Denemem ama hiç bir şey emin değilim (is this sentence correct)

Complete the missing parts in the sentences below.

1) Yaşlı kadının  evinde  beslediği   kedi  dün  onu  tırmaladı.

 The cat which the old woman feeds in her house scratched her yesterday.

2) Arkamızdaki  araba  bizi  geçmek  istiyor.

The car behind us wants to overtake us.

3) Bahçedeki   bazı  ağaçlar  yapraklarını  dökmeye  başladı.

Some trees in the garden have begun to shed their leaves.

4) Evinde  bir kedi  besleyen  yaşlı  kadın ağlayarak   bize  geldi.

The old woman who feeds a cat in her house came us by crying.

is it possible to say ağlaya ağlaya instead of ağlayarak? i guess they are same, but not sure

Absolutely. You can say "ağlaya ağlaya"  

How about ağladığı here, would it be grammatically correct, same meaning?

No, it wouldn´t be correct. It would have no sense if we apply -dık participle in there.

Unless we make the changes the sentence as below :

Evinde bir kedi besleyen yaşlı kadının ağladığı günü hatırlıyorum.

I remember the day when the old woman, who feeds a cat in her house, was crying.