Parts of this belong in the Danish WHS topic.... Anyhow.
3 Days Copenhagen (I assume full days):
* Helsingborg and Par force can easily be combined as parts of Par Force are on the same metro line as Helsingborg. I would plan max 2h for Helsingborg and 3h for par force hiking from one metro station to the previous one (see my review). So you should be back in the afternoon as could do some sightseeing of Copenhagen.
* Roskilde is a quick visit (1h, maybe 2h if you want to see the town), so again you should have time in the afternoon to do some Copenhagen sightseeing.
* Stevns Klint is a day trip if you use public transport. It takes two hours to get there. I had bad experiences with weekends, so be sure to check on google maps. Or better yet at http://www.dsb.dk
(hojerup kirke should be it). You should be back in the afternoon, so again, you have some time to see more of Copenhagen.
Copenhagen to Karlskrona to Kalmar to Visby to Stockholm
* Train from Copenhagen to Karlskrona (4h).
* Bus from Kalrskrona to Kalmar (1:30h).
* Bus from Kalmar to Oskarshamn and ferry to Visby.
Going from Kopenhagen via Karlskrona to Kalmar is feasible in a day withouth much hassle. Same applies for the trip via Visby to Stockholm, so I would skip the excursions from Öland (would be detours anyhow).
To check connections (dearly recommended, especially if you are travelling on the weekend, I would either use the national rail site of Sweden or Denmark. Or the German one (bahn.de). Bahn.de is pretty amazing and the Danish site imho is using the same software.
On your days in Stockholm:
* Birka (or more precisely Hövgarden) and Drottingholm can be combined in a day trip. I would first go to Hövgarden and then stop at Drottingholm. Both are on the same bus line.
* The mines (Falun and Engelsberg) imho can be combined in a strenuous day trip if you are travelling there during the week (no holiday, in summer, mon-fri). Falun is actually easier to reach as there is an occasional direct train connection. Historically, it's also the more important site from what I understand.
* The cemetery should easily give you half a day for sightseeing.
Imho, given the savings you could achieve by going to Karlskrona and Visby while in transit, I would consider adding a day to your Stockholm visit.