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‎‎1. I happened to see some dogs wandering around the street that looked severely wounded.

‎2. There used to be some dogs straying around the street that looked so scary.

3. There were many dogs around there that chased the deer

Q1) Are three sentences all correct English?

Q2) In each sentence, is "that" connected to "Some dogs", "Some dogs", and "many dogs"?

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fire1Q1) Are three sentences all correct English?

They're all OK (apart from the minor unrelated issue that one is missing a full stop).

fire1Q2) In each sentence, is "that" connected to "Some dogs", "Some dogs", and "many dogs"?

Essentially yes. (Whether strictly the relative clauses modify "dogs" or "some/many dogs" seems debatable.)

Comments  
fire1Q1) Are three sentences all correct English?

Close enough.

fire1‎‎1. I happened to see some dogs wandering around the street that looked severely wounded.

"Wounded" is usually reserved for soldiers' combat injuries. If the dogs had been shot or hit by shrapnel, then it would work, otherwise I'd go with "injured".

fire1‎2. There used to be some dogs straying around the street that looked so scary.

"So" sounds jejune in that use. Avoid it.

fire13. There were many dogs around there that chased the deer

You forgot the period at the end. I'm not sure you want the article in "the deer", but it works in a certain context.

fire1Q2) In each sentence, is "that" connected to "Some dogs", "Some dogs", and "many dogs"?

No, just "dogs", if you ask me.

 GPY's reply was promoted to an answer.