How tubes influence sound

Yes to all your questions. Tone is a biggy, upperfrequency emphasis can enlarge in width and depth the sound stage, and lower frequency emphasis can increase sense of warmth. Tubes can have a soft or rolled highs which might help reduce over-all system or source etch or brightness. Tubes brands and types both are different in sound from each other and in different applications. It is the knowledge of the differences and how to apply it is what makes the changing a tubes such an excellent way of tweeking system response to meet your tonal expectations.

Your question is quite broad-ranging, and there is really no simple one-fits-all answer. There are different circuit designs using tubes, different tubes and different utilization of the tubes. Some designers only use tubes in a portion of the circuit. A sort of historic answer is that tubes sound better in the midrange than SS and are not as extended at the frequency extremes. That is not necessarily fully true. I have found the most apparent differences to be in dimensionality/soundstaging, tonality, and harmonics or decay of notes. A good tube implementation will beat most SS units in those areas. These benefits may be more pronounced in the midrange, resulting in the reputation tubes have for better midrange. My current tube preamp improves on any SS unit I have heard in those areas, and is reasonably extended at both frequency extremes, provides good bass punch, and just simply sounds more real on vocals. Most of the tube preamps I have heard share similar characteristics, but like SS units some are better at certain things than others. It seems where they struggle the most in comparison is in the bass, where they do not always provide as much extension, definition and punch. You need to listen for yourself.

The answer to your 3 questions is 'yes'.

Different tubes can change the tonal balance, the level of detail retrieved, and the level of distortion added--and probably more.

Differences in tonal balance can make the system sound forward or recessed, brighter or darker (more or less treble energy), warmer or cooler, etc. Tubes can sound more or less detailed and more or less spacious.

Personally, I find these charactersitics to be subtle but real--but I'm not a person to use many superlatives as are SO many golden-eared audiofools*. Fortunately, there are at least several to dozens to hundreds of variants of the same tube type for us to choose from. UNFORTUNATELY, there are at least several to dozens to hundreds of variants of the same tube type for us to choose from.
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* Probably you've read that 'this makes a huge difference', or that 'it makes the difference between night and day', or my 'favorite'--'the sound is completely different'. To the latter, I respond--OK, the system sounded spacious before and now it's closed, constricted? The bass was tuneful and extended before and now it's...what? NOT extended? Not tuneful but is ill-defined? Well, you get the idea.
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if you start checking out online dealers in new/vintage tubes, they will often have fairly detailed descriptions of the tubes' characteristics....

The crux of it is stated in "A Taste of Tubes", see the link above by Jea48;

...the human hearing mechanism tends to interpret
low-order distortion as a welcome warmth and mellowness that can
enhance our enjoyment of music.

I realy enjoyed all the above descripition of what tubes can and will do especially in a pre.I also find this true in cd/phono/power amps, which is why many of us enjoy tube rolling...different equipment different tubes different presantation.This is one of the major reasons there is a demand for old stock tubes for they are still much better than todays new stock.