"She thought that he kept the keys to the house."
Translation:Hun troede at han beholdte nøglerne til huset.
What she thought really needs to be shifted back a tense in English - that he had kept...
I find really hard to understand when to use "at" or "det"...for instance, in this sentence I would have used "det"...
You only use det if 'that' refers to an actual object. (That house, that thought, or just 'that (thing)'.) Here it functions as a word that connects two sentences, so you need at.
According to Danske Ordbog, the past tense of beholde is "beholdt", not "beholdte". Beholdte is given as an informal conjugation. Why does DL prefer this one, is this form more temporary?
Yes, good eye. The verb "at holde" has two different past-tense forms, the standard holdt, and the more colloquial holdte. The latter is used frequently in the spoken language, and I'd say it's a bit easier to pronounce.
Beholde is derived from holde, but it officially only has the standard past-tense form beholdt. But due to the relation to holde, that informal past form is seeping into the standard speech. It's not correct (yet) to say or write beholdte, but it's likely to become more accepted as time goes on. The same is true for other derivatives of holde, such as indeholde, udholde, or forholde.