omission of -ing

1) Dhaka standing by the side of Buringanga (a river of Bangladesh) is the capital of Bangladesh.

I can say this sentence in the following way.

2) Dhaka that stands by the side of Buriganga is the capital of Bangladesh.

In case of first sentence, since 'that' has been omitted, 'standing' has been used instead of 'stands'. Am I right?

Let consider the following sentence.

3) He found a toy that was under the chair.

I can say it in the following way.

4) He found a toy under the chair.

a) In case of fourth example, 'that' has been omitted. But why 'being' is not written instead of 'was'?

b) Would you like to explain when '-ing' is used if relative pronoun is omitted, and '-ing' is not used though relative pronoun is omitted?