Because your husband is not French, you should be applying for a Carte de séjour "de membre de la famille d'un citoyen de l'Union/EEE/Suisse".
It's not clear to me why they would care about the proof of your whereabouts, but there is a requirement to apply for the card within 90 days of your arrival in France. If you apply after 90 days, the cost rises from €25 to €340. If you apply within 90 days but they drag their feet, the cost should remain €25, so, because you moved in October and applied in October, you should be paying the lower fee. There is no other penalty for a late application; in particular, you cannot be deported.
We traveled inside Europe from the time we arrived in the summer of 2015 until we moved into our new house in France in October of 2016 (15 months). I did not secure a residency permit anywhere, because we were staying with friends and didn't really have a permanent residence. Perhaps I should have applied for the CDS when we signed for the house in January 2016, even though we really weren't staying in France for extended periods. Just trying to find out if it's going be surmountable after all this time.
Yes, it will be surmountable. As mentioned, the only real question is whether to charge you €25 or €340. I do not know exactly what criteria they would use to assert that you owe the higher fee, however. If you were never in France for more than 90 days, then they probably cannot charge the higher fee. If you were in France for more than 90 days, uninterrupted, before filing your application, then they certainly can charge you the higher fee. If you were in France for more than 90 days, and then left, and then moved to France, they might argue that the higher fee is due, and although I doubt that this would stand up in court, the cost of a court battle is not justified by the €315 difference.
The 90 day period is triggered by your arrival in France, not your purchase of property. If you were based in France for your 15-month travels in Europe, then you should arguably have applied for the card then; a short trip away from France would generally be held not to interrupt your period of residence. On the other hand, if you were in France for just a few days, but spent most of your time in Germany or indeed anywhere else in the world, it would be difficult to argue that you had established yourself in France at that point or that your period of residence should be said to have started then. In any event, their interest in your whereabouts seems consistent with an inquiry into whether to charge the higher fee, but whether that is the real reason or not is still unclear.
It seems, however, that your major worry is whether they can deny the carte de séjour altogether. In general, they cannot.
More precisely, they can deny it only on very limited grounds, which almost certainly do not apply. The grounds are:
- Your husband is not an EU citizen
- Your marriage to your husband is not valid
- You and your husband, as "economically inactive" people, are not able to support yourselves
- You or your husband are a danger to public health
- You or your husband are a danger to public safety
- Your or your husband's presence in France is contrary to public policy
The third item, being able to support yourselves, is only necessary for the first five years; after that time, you gain a right of permanent residence in France.