I too suggest Wikipedia as it has a lot of the info you wanted.
Diana had two older sisters and a younger brother. Also a stillborn brother born before her. Her mother was under a lot of pressure to produce a son and heir and Diana's birth was a great disappointment apparently. Not the best start in life despite her wealth. Her parents had a messy divorce when she was still young (her mother left the family which was usual in the aristocracy as the father would just hire nannies etc) and both remarried.
She is buried on an island surrounded by a lake in the grounds of Althorp, her ancestral home. There were rumours that in fact she isn't buried there as the ground is too marshy and she is actually buried in the vault of Althorp Church in the village but this was kept quiet to avoid it becoming a pilgrimage place for Diana fans. Some brick and stonework was carried out there the week of her funeral but no idea if any truth in the rumour.
She didn't have a state funeral but it was a very grand/public ceremonial one in Westminster Abbey - televised live. I have to say until I watched the queen's funeral this week I wouldn't have known the difference between state v ceremonial.
As another poster has said her position/ title on divorce was amended to Diana Princess of Wales (not HRH). I imagine she would have retained that for the rest of her life unless she remarried. Then she would have reverted to Lady Diana (her born title). It would have been quite awkward for Charles to remarry if Diana was still alive as there would technically have been two Princesses of Wales, even though Camilla never used the title. They (charles and camilla) had to have a civil wedding ceremony because the official position of the church of England is not to marry parties who still have a living former spouse. By that time Diana was conveniently deceased but Camilla's former husband was (and still is) alive. Otherwise they could have had a full church ceremony despite their earlier divorces.
If Diana had lived I don't think she would have remained as iconic. The tragedy of dying young and suddenly helped retain that image. I think she may have remarried, possibly had more children and possibly moved abroad once public interest in her waned. It would have been nice to see her in a happy marriage. She and Charles barely knew each other and were never compatible. She was chosen for him because she ticked all the right boxes including never having a boyfriend, being a titled lady etc.
By all accounts she was a warm affectionate and fun mother but she did spend a lot of time away from her kids. They went to boarding school and she travelled a lot (sometimes with them but more often without). That's not to say she wasn't a good mother but the lives of the aristocracy are so different to ours. She definitely wasn't "one of us". Yes she wore casual clothes when off duty but so do all the royals. They weren't from Primark though!
I would have been very surprised if Charles had mentioned her in his speech. Most people would not have considered it appropriate. Talking about the love and pride he has for his sons was as close as it got.
I don't agree she was universally adored and admired. She wasn't perfect (who is?) and many people thought her courting the press and "Queen of your hearts" stuff was cringey and manipulative. She was not very bright (which she freely admitted) and traded on her looks and touch-feely compassion. Many people absolutely loved that though. Again I wonder if she would have reduced in popularity as her youth and beauty faded.
I don't think she was murdered. It was quite convenient but they were already divorced. It would have made more sense to bump her off earlier and avoid the public separation and divorce. She was already "out" of the family and doing her own thing. If she had been murdered it would have been done much more discreetly, not with the three potential witnesses plus a pack of press in hot pursuit. The driver was drunk and speeding and only one person was wearing a seat belt (who survived). A horrible accident.