So what's the difference between those two words? Is bathtub just the object that holds the water, whereas bath can mean the facility and also the process?
Yes, and a bath can also be the water in the bathtub, or perhaps both the water and the tub together as a collective entity.
Is there a stylistic preference for saying "in the bathTUB" as opposed to "In the bath"?
As a native speaker from California, I would say "she is in the bath" if she were taking a bath. I would say "she is in the bathtub" if she were merely lying or standing in the tub. For example, if a little kid playing hide-and-seek climbed in an empty tub to hide (and I were so mean as to give away her hiding place), I would say "she is in the tub."
I would say "she is in the bathtub" if she were merely lying or standing in the tub.
I would say, "she is a strange one, that girl of ours."
As a British English speaker, I use "bath" for the modern plumbed-in equipment but would use "tub" or "bathtub" for antique or portable baths
Sometimes the audio does not sound correct. Here on normal speed it sounds like there is an extra syllable in 'baignoire' -> Baign-i-oire
Is this happening to others too?
I think it's because the "gn" in french is usually pronounced as "nya". (Magnifique sounds like "Manyafeeck".) And when there's a vowel after the "gn", the pronunciation needs to be broken down to accommodate the sound of that vowel, hence the (extra) syllable.
A liaison question:Should the est after sœur have an added r sound in the beginning. And if it should why?
The "r" sound is there at the end of "sœur" regardless of what comes after it. This isn't a case where liaison makes a latent sound appear that would otherwise remain hidden.
However, you could say that as an example of enchaînement or resyllabification, the "r" sound does move to the beginning of "est".
But this idea can be applied to English too. When I taught English to Chinese speakers who had trouble pronouncing certain terminal consonants, I suggested that they imagine the terminal consonant in question not as the ending of a word but as the beginning of the next word in the sentence, and magically they could then pronounce it correctly.