In my opinion, 1 and 2 don't sound idiomatic while 3 definitely does. However, I'd imagine one would be much more likely to say "the room looks clean" rather than "the room feels clean" or perhaps "the room feels nice and clean."
I'm just basing those examples on what I think you're more likely to hear in daily speech.
For number 2 (if you really had to word it using that structure) I'd change it to:
"The room gives off a feeling of cleanliness" to make it sound less "off."
I'd say that there is nothing wrong with 1, it just sounds a bit more "proper" than 2 and 3.
I found an example online using the same wording as 1:
"On the whole the room gives an impression of bareness, but also an impression of cleanliness and neatness."
As for all 3 sentences, they all suggest the same meaning.
Thank you very much for the very helpful answer, hjr-lm7mudia-hntati. Good to learn the version with "off".
I'm also wondering if it's possible to reverse the structure of sentences 1 and 2.
3a below is a slightly modified version of 3, and I reversed the sentence to make 3b.
3a. The room feels/looks clean because of the beautiful carpet.
3b. The beautiful carpet makes the room feel/look clean.
Would "The beautiful carpet gives the room an impression/a feeling of cleanliness" work? I think "impression" doesn't work in this case.