I feel "There appear + Noun" is rarely used and might sound strange.

You are incorrect.

Even though that phrase is not rarely used, I would say that by convention "There appeared ..." is generally followed be "to be". I think the text is not perfectly translated and that the correct way of writing these particular phrases that you highlighted in bold are as follows:

if there appeared to be certain influence

and

There appeared to be no sign

respectively.

Examples

However, this does not mean that "There appeared + Noun/Adjective" is rarely used, nor does it necessarily sound strange when it is used correctly. Bear in mind these valid examples (the first two were also mentioned in the comments):

There appeared in the doorway a genie.

There appeared in the doorway a tall, menacing highwayman.

I appeared as if from nowhere.

It would appear that there was no one at the front door.

There appeared, somehow, to be another room in the house.

Bottom Line

"There appeared + Noun/Adjective" can be used correctly in certain circumstances, but most of the time it tends to be followed by "to be".

The single word you may be looking for is

lull - calm or send to sleep, typically with soothing sounds or movements

I'll sing a song to lull you to sleep.

as in lullaby.