2 weeks of vacation after 6 month work stint in Germany

I would consider the Pulglia region of Italy. The beach areas should be nice in early October. Plus there are wonderful sites in Lecce and Matera (in the providence to the west).

Your wife can not stay for the 3 months with you PLUS the two weeks afterwards if she doesn't get a visa.

First of all the time she can stay in the Schengen zone (most European countries, see here for a list - https://www.axa-schengen.com/en/countries-schengen-area ), is 90 days, not 3 months. You can only be in those European countries for 90 days out of any 180. So by the time you finished up your TDY, she would have to leave.

If you want her to travel with you when you're done, you'll need to calculate BACK from your planned departure date from Europe to see on what date (90 days before) she can arrive in Europe and stay legally without a visa.

(Edit to add: Exception to this is if you spend your last two weeks after the 90 days in the United Kingdom or Ireland, since they're not part of the Schengen accord.)

2 weeks vacation in October and something completely different than Germany. Beach and relaxation. I am not sure about beaches in Croatia in October. I would look at Greek islands (in Santorini I swam in the sea at the beginning of November and it was somewhat cold but still swimmable). Or southern Spain or even better Canary Islands (is it Europe or Africa? Well it is the part of Spain). I liked the Tenerife Island the most. Very geographically diverse. It is enough to the south that the ocean is swimmable all year long.

Greece was the first place I thought of but also Sicily, Puglia, Croatia, The southern coast of Crete is beautiful and the water is even warmer there.

Maybe start in Naples, then head south discovering beaches of Campania and Calabria, before finishing on the east coast of Sicily.
Scilla and Tropea on the Calabrian Coast are 2 places that have my interest at the moment.

Much of Croatia, other than the bigger resorts will have (or soon will be) closing up for the winter once you get into October and there is a fair chance that the good weather will have broken too.

Greece can be ok but I have also experienced dreadful weather there in early October.

For a better chance of good weather, I would head to southern Spain - Seville and Cadiz or Jerez would be a good mix - or to Portugal - say Tavira in the Algarve combined with either Lisbon or Seville.

If you want the closest thing in Europe to guaranteed sunshine in October, I would suggest the Canary Islands - Fuerteventura has some of the best beaches and is one of the least developed. There are many flights there from Germany.

When checking your Schengen time, don't forget that both the arrival and departure days count.

Checking the Schengen list, I see that Croatia, Albania, Serbia, Montenegro, Bosnia-Herzogovenia, and Macedonia are omitted. So, go to those. We spent 2 weeks in Oct-Nov in 2014 in Bosnia, Serbia, Montenegro, Croatia. It was cool-cold in Sarajevo, warmish in Dubrovnik, pleasant in Mostar, pleasant in Novi Sad/Beograd. Dubrovnik in particular was quite nice, because the hordes of tourists were not there. I recommend Croatia and the Balkans for both off-the-beaten-trackishness and for cost. It's hard to beat Croatia/Serbia now, and in 10 years, things will cost more. Plus these are more slavic than germanic, and you will get a different cultural vibe. Turkey is also an option, as it is not on the Schengen list either. Note that if your wife stays in Germany for 90 days, you will need to leave directly from a non-Schengen area. You may wish to consider Turkish Air, and spending a few days in Istanbul. It's a crazy place.

Consider the big three cities in Italy, Venice, Florence and Rome. You can do these places well in 2 weeks.
Another choice is Spain. Madrid, Toledo, Segovia, then Seville.

Bulgaria and Romania are also non-Schengen.