Why does Arthur Clennam suspect Blandois, and of what?
As readers we know that Rigaud / Blandois is probably a villain. But Clennam has had no contact with him and knows nothing about him, yet takes strongly against him and threatens to throw him out of his mother's house almost on sight ("If I were the master of this house, I would lose no time in placing you on the outside of it."), apparently just because he's seen Blandois earlier talking to Miss Wade and Tattycoram, and he looks a bit foreign.
At the time Dickens was writing, that's about all it would take. In his earlier appearances Dickens has been at great pains to indicate that Blandois looks and acts the English stereotype of the untrustworthy foreigner. Taken together with the other common stereotype of good characters being instinctively able to discern evil (note that both Little Dorrit and Pet Gowan also distrust Blandois, for even less reason) and you've got one characteristic Dickens flourish. Genius the man indisputably was; subtle he was not.
In the BBC version, Arthur sees Blandois "shushing" Tattycoram with a condescending smirk and a finger to her lips. Since she left the Meagleses to get away from that sort of treatment, putting up with it now means she must be seriously scared.
What's up with Blandois's "disappearance"?
Why does Clennam (or possibly his mother, it's not clear) suddenly decide that Blandois has vanished, and devote time to trying to track him down? Apart from seeing him with Miss Wade, the entire extent of his dealings with the man is to bump into him at Clennam's mother's house, which Clennam knows he has visited once before on "business". Why would he know or care where the character went after that, let alone send out handbills and even travel to France looking for him? What's he trying to achieve?
He doesn't send out the handbills, Mrs. Clennam and Flintwich do, and they have an excellent reason for wanting to find him — they need to know how much he knows, and whom he might be planning to share it with. Arthur's motives for joining the search seem to be mostly about his subsequent natural curiosity as to what's going on with his mother and her association with the man. Bear in mind that even giving Blandois the time of day is way, way out of character for Mrs. Clennam, and that her son already has strong reason to believe she's hiding disreputable secrets.
When he does find Blandois, Clennam accuses him of placing his family under suspicion of murder. This seems to be based on Clennam's own, somewhat paranoid idea previously that Mr. Flintwich might have killed Blandois, by order of Mrs Clennam. But why would anybody else possibly think that Blandois had been murdered? Nobody but Mrs Clennam and Flintwich knows of the blackmail at this point. No other character, much less any authority, has reason to know of Blandois's existence, or to connect him with the Clennams, let alone to think that he's "disappeared". Even as readers, we don't know why Arthur Clennam thinks he's disappeared. What's more, Blandois himself seems to agree with Clennam's interpretation, and says that he was in hiding because he thought that it would be funny to create this suspicion.
And the "disappearance" makes no sense anyway, even if creating unfounded suspicions was Blandois's motive. Blandois is blackmailing Mrs Clennam, and needs to reappear to set up the last meeting and try to claim his money anyway. If anybody - for whatever reason - had somehow thought he was dead, at that point he would have turned up alive and the whole thing would have been dropped. It simply doesn't go anywhere in plot terms.
In character terms, though, it makes quite a lot of sense, as per the note above about pulling the stunt because he finds it funny. Blandois isn't just about blackmailing this specific family, he's generally out to take revenge on a world that refuses to treat him as the gentleman he considers himself to be. From that angle, his making this old, established family run in distracted circles on his whim is entirely plausible.
It's still pretty lousy plotting.
Why won't Mr. Meagles call "Tattycoram" Harriet when he finds out she hates the nickname?
Tattycoram, or Harriet as she prefers to be known, runs off to live with Miss Wade. When Clennam and Mr. Meagles track her down and she explains why she was unhappy, one of her complaints is that she hates the name Tattycoram because she finds it demeaning. Which (together with her other points about being treated as a second-class family member who's wheeled out to prove their benevolence in adopting her, rather than treated as a real daughter) seems perfectly fair based on what we've read so far. But rather than apologise and at least say that he didn't know she hated the name, and that he'd call her "Harriet" in future if she came back, Mr Meagles has a long rant about how ungrateful she is, and how he's going to carry on calling her Tattycoram whether she likes it or not because he means it affectionately and she should appreciate that, and so on. Clennam backs him up. Rather than Meagles having a change of heart, by the end of the book "Tattycoram" has realised the error of her ways and come back under the same terms, and we're apparently supposed to take it as a lesson in ingratitude. Surely Dickens usually has more sympathy than that? He can clearly see things from Harriet's point of view since he wrote her complaints in the first place.
This troper suspects that it has something to do with Values Dissonance, specifically the class systems of the time. Perhaps Dickens' idea was that as the Meagles were upper class, they knew better than Tattycoram/Harriet about how to raise her and that she was lucky at all for them to have taken her in. It doesn't make much sense, but apparently Dickens was very class-oriented. An example would be why Ester refused to marry Mr. Guppy in Bleak House - she was technically still the daughter of a noblewoman, and thus Dickens wouldn't have her marry a man beneath her socially.
Indeed, that makes some sense. But it's exactly the kind of social stricture that I (the previous troper) would have expected Dickens to be against. Guppy was an idiot, Esther wasn't - she couldn't have happily married him anyway. Not the same. (And she didn't know her position...)
Dickens was against it, or at least not completely sympathetic. Note that by this point Meagles has been established as an unmitigated snob — when his precious daughter marries an obvious cad, he comforts himself with the fact that the man at least has good family connections. The point behind their final scene isn't so much that Meagles is right, as it is that Tattycoram has learned to submit to her 'duty' regardless — ie. to be patient, forgiving, loyal etc. etc just like Little Dorrit, whom Tatty's explicitly supposed to take as her model. It's still terribly awkward to modern readers, but not unsupported by the author or his text.
Why do people devote so much time to looking into the Dorrits' history, for no obvious reason?
Arthur Clennam decides on sight that his vague suspicions that there's something wrong with the family business must clearly have something to do with Little Dorrit. (He's right, of course, but even the readers haven't any real reason to think this is the case until close to the end of the book). So he takes it on himself to look into the family affairs at the Circumlocution Office. To be fair he gives up quickly, in line with his complete ineffectualness in most of the rest of the book. But Pancks also has exactly the same instinct, again simply from seeing Amy Dorrit, and devotes all his free time for months and even spends his entire personal savings on discovering and sorting out Mr Dorrit's affairs. In Pancks's case he apparently makes a vague connection between the surname and an unclaimed inheritance he's heard of, but it's lucky for him that he's right or he'd have wasted all his time and money on a random impulse. It's as if they both know they're in a Dickens novel called Little Dorrit.
Arthur knows that it would be very unusual for his mother to employ someone like Amy, and show her the kindness that we see. Therefore he probably guesses that there must be some sort of connection between the families, and decides to look into it.
True. It's also worth noting that some little time is devoted to establishing Pancks as exactly the type of man who a)really enjoys this type of financial investigation and b)will not let something go once he's well started. These same traits are in evidence again near the end, when Merdle's bank crashes and both Pancks and Arthur (the latter acting on the former's advice) lose everything — Pancks is devastated that he should be so wrong re: money matters, to the point where he takes comfort in redoing his calculations over and over to prove that the investment should have been a sure thing.
Why did Clennam's uncle randomly decide to leave money to Amy Dorrit?
OK, Clennam's great-uncle decides in guilt and compassion to leave money to Clennam's real mother, to repair the damage done when Clennam's father was forced to abandon her for Mrs Clennam. This makes sense. But why would he also add a clause in the will that leaves money to the youngest niece of the patron of Clennam's mother? (Which of course turns out to be Little Dorrit). Particularly as he doesn't even seem to know much about the family of the patron - Frederick Dorrit, Amy's uncle - because there's a clause that if he happens to have a daughter of his own at 50, then the money would go to her. It says this is some token of disinterested appreciation of the patron's efforts, but that seems a real stretch.