1. It'll take thirty minutes for her to get here.

2. It'll take her thirty minutes to get here.

(Are both the same and grammatical?)

3. As it rained heavily, the playground is muddy now.

4. It having rained heavily, the playground is muddy now.
(Are both the same?)

5. As it rained heavily, the playground was muddy.

6. It raining heavily, the playground is muddy now.
(Are both the same?)

1 and 2 are the same, yes.

3 & 5 are OK, but 4 & 6 are not correct. It'd be better to write this:
After heavy rains, the playground was muddy.

and

After hours of heavy rain, the playground is muddy now.

As it rained heavily, the playground was muddy.

1. Yes, both sentences are grammatically correct and convey the same meaning. In both sentences, "thirty minutes" is the amount of time it will take for her to arrive at the specified location.

2. In this sentence, "her" is the subject and "thirty minutes" is the duration or time it will take for her to arrive.

3. Both sentences have a similar meaning, but there is a difference in terms of verb tenses. In the first sentence, "As it rained heavily," indicates that the heavy rain happened in the past and has concluded, resulting in the playground now being muddy. The second sentence, "It having rained heavily," uses a present participle phrase to describe the current condition of the playground based on the past event of heavy rain.

4. The first sentence, "As it rained heavily, the playground was muddy," uses the past tense to describe the condition of the playground as a result of the heavy rain. The second sentence, "It raining heavily, the playground is muddy now," uses the present progressive tense to describe an ongoing action of heavy rain resulting in the current muddy condition of the playground.

In summary, while the sentences may have similar meanings, there are differences in verb tenses and structures that convey slightly different nuances.