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.
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?)
2 answers
As it rained heavily, the playground was muddy.