Yes! Thats very much how I see her childhood.
John Faa makes a comment when he's telling Lyra about her parents for the first time, something about Mrs. Coulter being "not so well-born" as Lord Asriel. I don't know how the English system of titles goes, but I'm fairly sure that Lord Asriel would've inherited the title of Lord from his father. So I'm guessing his parents were some pretty powerful, not to mention some very rich, people.
1) The original story was that Lyra's father was a Count and he and her mother had perished in accident, leaving her an orphan under the care of her uncle, Lord Asriel. Count is not a British title, the equivalent is Earl (from the Middle English "Eorl" or "Warrior")., which indicates that the Belacqua family is not originally English.
2) As Lyra's "father" held a title, it indicated that he was the eldest brother. If the grandfather was dead, he inherited that title. If the grandfather was alive, the eldest son would have assumed the next highest family title. (ie: The son of Duke "ABC" might be Count "XYZ" until his father dies, when he becomes Duke "XYZ" and his son becomes a count)
Asriel, as a younger son, would have the courtesy title of Lord Asriel
3) As Lord Asriel did not become Count Belacqua upon his brother's death, there must have been a middle brother who inherited the title of count.
So Lord Asriel was the scion of Continental nobility - yeah, they probably had a few (insert local currency) to throw around.
By the way, as far as Lyra goes -
1) Under the Lex Salica (which applies to a good portion of the contenent), she could not inherit the title. It went to the nearest male relative, no matter how distant.
2) If the world never discovered he truw parentage and it was still believed she was the daughter of a duke, she would be entitled - in Britain - to be addressed as "Lady Lyra"
3) If Asriel acknowledged her as his daughter (say, in a will), as the daughter of a younger son she remains "Miss Belacqua"