Easily doable.

As a previous poster indicated intra-Europe flights can be found for very cheap rates on carriers like Wizz Air, German Wings, Ryan Air, Brussels Airlines, Air Baltic etc. if done in advance

I'd recommend something different from the previous posters though. I'd get the cheapest round trip to Europe. Checking May I found $600 including taxes to both Amerstdam and Brussels (direct from JFK).

Then I would rent a car for a week and see Belgium & Southwest Germany Black Forest area. Best Western is a good solid hotel chain in Germany and rates include breakfast. Depending on the dates you can find rates between $60-$100 Euros - figure 6 nights 2 rooms sets you back 1100 euros plus a week car rental at $500 euros. Then I'd try to find a $100 plane ticket to Helsinki and spend three days there (if you have hotel points I'd use them in Helsinki as its expensive). From the airport there is a finnair bus for about 5 euros into the center of town. $300 euros for plane tickets and another $100 for the ferry.

Then I'd take the ferry to Tallinn. And I'd stay 2-3 days in Estonia, 2-3 days Riga Latvia (via train) and a day or two in Vilnius (again via train). You can usually find decent hotels like Scandics for less than 100 euros in these cities if you book in advance and they include a good breakfast. Another $1200 euros in hotels and figure $200 euros train tickets.

Then I'd find the cheapest airfare to either Vienna or Bratislava and rent a car and spend 3-4 days in Slovakia visiting castles, old towns and hiking through beautiful nature - and either swing down to Budapest or spend the final few days in Vienna (which is admitted not cheap but has great public transport). Then fly out of Vienna back either Amsterdam/Brussels and next day go home. Another $500 euros car rental plus 6 nights hotels at another $1500 euros (again use points if stay in Vienna).

Thats a total of $5400 euros ($7800 USD) in Europe plus $1800 in airfare. So if you then allow $150/day food to spend for 21 days thats another $3150. Total of 12,750 and $2K to spend as you see fit.

If you have various points use them on hotels in big European cities. If you have to see Prague then use points. If you have to see Vienna again use points. Don't associate Best Westerns in the US with Best Westerns in EU. You'll find that small hotels in Europe are often of good quality.