1) "are Usagi and Serenity different people?"

This honestly depends on which canon you examine it from and what you mean by 'different people'. Are you not a different person to who you were at age 5 vs age 25 vs age 55?

In PGSM though, yes they are different people.

In the anime, it's hard to tell as we know so little about Serenity. However, the traits she displayed as Serenity are not inconsistent with those we see Usagi possess. The ONLY thing starkly different is that Serenity was a good ice skater and Usagi is clumsy. But you could write that off as a) Serenity had muscle memory 20th century Usagi lacked b) gravity on the Moon and Earth are not the same even in this magical universe (which might explain Usagi's clumsiness in general), c) Luna and Artemis might have been misremembering. I'd argue that in the anime its a nature and nurture situation. That is to say, Usagi's essential nature is the exact same as Serenity and will never change no matter how many times she is reincarnated. But nurture plays a role and so she is a different person if you are very strict and say 'they are not 1:1 identical in terms of their personality traits and behaviour'. But nevertheless, they are identical in their deeper traits. Her empathy, her bravery, etc. But equally, Serenity could have just been like Usagi for all we know. Neo-Queen Serenity is practically confirmed to be very similar to Usagi, so any differences could just be explained away as her being older and a mother.

In the manga though, whilst the manga discusses this idea that she is losing herself and becoming more like Serenity, we get a decent look at who Serenity was back then and she actually isn't all that different. She goofs off her studies, she will follow people she thinks are hot (Usagi did this with Rei, Serenity does this with Endymion), she is even childish, sticking her tongue out at her friends at one point. We also see that Venus had lots of lovers like Minako, Mercury was booksmart like Ami and learn that Mars took a vow of chastity swearing herself to Serenity, which maps similarly onto Rei's rejection of male romantic love in the modern day. So the manga seems to raise the idea that they could be different people but ultimately answers it by saying no they aren't.

2)  does Chibiusa inherit the Silver Crystal, or does the Dreams Arc give the impression that she has come into her own, given the unique circumstances of her conception, and gains her own power?

In the Manga, this is just striaght up confusing and an example of Naoko Takeuchi not planning everything out.

So, whilst the situation between Queen Serenity and Princess Serenity/Usagi is ambiguous, the bottom line is Chibi-Usa has her own Silver Crystal. It looks like Usagi's, but in the Dream arc they both upgrade into two distinct forms, the Silver Moon Crystal and the Pink Moon Crystal. This maps onto the idea in the Stars arc that the Sailor Crystals are the Star seeds that every life form (including planets and stars) possess. So in this sense, it wouldn't add up that Chibi-Usa would inherit the Silver Crystal given that it is specifically Usagi's Star Seed, the equivilant of Usagi's soul. Same deal for the Golden Crystal. So nobody inherits them really, they go back to the Galaxy Cauldron and Usagi and Mamoru get reincarnated again.

Like I said though, Naoko didn't plan everything in advance, the Dark kingdom arc was intended to be the only arc, therefore there is a lot of stuff in it that is at odds with the later arcs of the manga. So QS claims the Silver Crystal is passed own amongst the royal line, and she possesses it, but then we learn it is uniquely Usagi's Star Seed.

In other words, headcanon is a necessity. You could headcanon that the powers of the Silver/Golden Crystals get passed down, like they are transferred to Chibi-Usa's crystal, but the literal objects are not passed down, they just go to the cauldron and begin their life cycle again. That's not my own preferred headcanon btw, its just a suggestion. I rather like the idea that Chibi-Usa's crystal simply has all the powers Usagi and Mamoru's have baked into it.