Since it's a bit too long to answer in comments, I will put my previous comment along with additional information here.
They all look generally okay, in my opinion. But I would prefer "rapidly flowing river" or "fast-flowing river" instead. (I might reduce that to "rapid water" or "rapid river", maybe.) Also, "playing happily" sounds better than "happily playing". (That's only my opinion. I think you can use either.)
Trying to put all those into one single sentence, I personally would go with "Sitting on the bridge over the rapidly flowing river, I watched the children playing happily in the first snow of the season." (If this is the first sentence of the passage, I might change "the rapidly flowing river" to "a rapidly flowing river", unless the river itself is important to the story.)
As for another question in your comment, "If there is no "were" (helping verb), I have to move adverb to stay after verb (playing)?" made me think. Generally speaking, English is quite flexible on adverb position, and I simply write the way I am familiar with. However, to answer your question, I checked out a grammar book (which also confirms that both "playing happily" and "happily playing" are acceptable), and found many entries on the usage. This one is perhaps the most relevant to your question,
24 adverb position (4): mid-position
7 adverbs of manner
These adverbs say how something happens or is done.
Examples: angrily, happily, slowly, suddenly, noisily, quietly, softly.
Adverbs of manner most often go in end position (see 23), but adverbs ending in -ly can often go in mid-position if the adverb is not the main focus of the message.
She angrily tore up the letter.
I slowly began to feel better again.
We have suddenly decided to sell the house.
This time next week I'll be happily working in my garden.
Mid-position (after all auxiliary verbs) is especially common with passive verbs.
The driver has been seriously injured.
Practical English Usage by Michael Swan
The part "Adverbs of manner most often go in end position" explains my preference of "playing happily" quite well. But still both positions are allowed, and according to the book, if the adverb is not the main focus, it often goes in mid-position.