Why not "thuigeann tu"?
because the structure is "can you understand..." no "do you understand..."
Is it possible to say "an féidir leat na Gaeilge a thuiscint?" here? Instead of "an féidir leat Gaeilge a thuiscint?" the first wasn't accepted.
Nope, because Gaeilge isn't a plural. You can use an Ghaeilge, however, which is what's shown as the translation here.
Go raibh maith agat :)
What is the difference between an Ghaeilge and na Gaeilge, and as Gaeilge?
an Ghaeilge is the nominative, when speaking abouy the language in a broad sense. na Gaeilgej is genitive. as Gaeilge is when you speak as Gaelige*
Duolingo accepts both Gaeilge and an Ghaeilge here. Should it accept the indefinite version? I gathered from another sentence's discussion that the indefinite is often used, but is technically incorrect here?
Why can't you say, "An bhfuil tú an Ghaeilge a thuiscint"?
Because it doesn't make any sense - "Are you to understand Irish?".
You could say An bhfuil tú in ann an Ghaeilge a thuiscint? or An bhfuil tú ábalta an Ghaeilge a thuiscint?