The Weasleys are incomparable as far as families go in the Harry Potter universe. Beyond Ron being best friends with "The Boy Who Lived" the family is also unique in its size, its history (Gryffindors, every single one), and its kindness. No one deserved a Christmas sweater more than Harry and Mrs. Weasley made sure he got one.
However, just because the Weasleys are loved does not mean they are perfect, or even that they always make sense in the series. The existence of the Weasleys actually raises a lot of questions about how they were able to manage, survive, and be as happy as they were.
Had Seven Kids
The Weasleys were poor. This is obvious from the beginning when Ron appears with his wrapped sandwiches and his hand me down robe. This isn't a character failing, it's just a fact.
However, it is a bit strange that, knowing their economic situation, Arthur and Molly decided to have seven children. There's no rule about how many children they were allowed, and if they wanted more than one, more power to them. But seven isn't just a big number where children are concerned, it's huge.
They Are Pure Bloods
It makes sense that the Weasleys were a pureblood family, what is interesting is that they are still a pure-blood family by the time Harry meets them.
They've been blood traitors and fought in the First Wizarding War and yet, despite marriages across the family, they still hold their "pure blood" status. It may have only been broken when Ron married the muggle-born Hermione after Deathly Hallows. It doesn't really make sense that the family's pure-blood status was intact for so long.
Don't Communicate More
With seven children all growing up together out in the country it's strange that the family isn't better at communicating with one another. Fred and George don't tell Ron about the Marauders Map.
Ginny goes into a depression and can only pour her heart out to an enchanted diary. Percy writing to Ron in his fifth year isn't expected, it's a surprise. For such a big and (supposedly) close-knit family, they don't really share important information with one another.
Pettigrew
This is specific to Fred and George Weasley, but it's still a major question. How did Fred and George not notice Pettigrew's presence on the Marauders Map?
More importantly, for the Weasleys, how did they not notice Pettigrew always being with Ron? Including in his bed? Why didn't they investigate, or just ask Ron. It doesn't seem like they were being very good protective older brothers.
Arthur's Job
Mrs. Weasley doesn't work, so the family is completely supported by Arthur's government salary, and it's clear that this is not enough. However, given the prominence of the Ministery in the Wizarding World and the overall small community of witches and wizards. Shouldn't the magical community be making sure all of its members, including the Weasleys, are flourishing?
While this isn't the Weasleys' fault, it's just something that doesn't make sense.
More Chores
In Harry Potter and the Chamber of Secrets Ron, Harry, Fred, and George have to get gnomes out of the garden, otherwise we don't see the kids doing many chores. Mrs. Weasley seems to take care of the house and the family single handed, which doesn't seem quite fair. One would imagine that the children would have more chores when home at the Burrow, but it's never made clear on the page or on the screen.
To keep everyone happy and things running smoothly it would make sense for there to be more assistance from the young ones.
No One Worries About Percy
Percy is smart, he makes Head Boy after all, but he is also distinctly different from the other children in his family. He stands out. And, even so, no one seems to worry about him.
It's hard, understandably, with so many children, and so many issues (including Ginny being possessed and Ron almost dying year after year) but the fact that everyone is surprised when Percy has a falling out with the family doesn't make sense. Someone should have noticed him and worried a bit about him much sooner.
Ron Keeps Going to Hogwarts
In his first year Ron helps fight off Voldemort. In his second year, Ron faces an acromantula and a basilisk. In his third year, Ron has an escaped murderer standing over his bed.
How Molly and Arthur continue to convince themselves that Hogwarts is a safe and important place for their youngest son to live and learn is very hard to understand. They've seen five other children go to the school and no one has had an experience even close to Ron's. This isn't normal.
Homeschooling
At one point Molly was in charge of homeschooling five of her seven children at one time. How she was able to do this single-handed (as Arthur was off working at the Ministry of Magic) is hard to fathom.
Molly was always talented, very good at juggling tasks, but, with Fred and George in the mix of children at home, it doesn't really make sense that she was able to manage. Perhaps she was able to keep them all safe, but how they got any satisfactory education out of the experience is questionable.
Understanding of Muggles
Mr. Weasley is is charge of the Misuse of Muggle Artifacts Office at the Ministry of Magic. That he is in charge of this office but doesn't really understand the muggle world is strange.
How can he identify when a muggle artifact is being misused if he doesn't really know what it does? Muggle studies must not be a very good department at Hogwarts.