Starting with Sofia, we also plan on exploring Plovdiv, Rila
Monastery, and Veliko Tarnovo.
Those are good towns but with a car you should also be focusing on the countryside. Around VT, google Etar, Kazanlak, Arbanassi, Russian Shipka Memorial Church, Dryanovo Monestary, Buzludzha, Kazanluk and the Madara Horseman, all of which could be done as side trips. I pretty much always visit VT and I love this town, it gets better every year.
Plovdiv itself is fantastic, stay in Kapana or the old town if possible, there are plenty of places with parking. From Plovdiv, I would recommend taking a southern loop through the rhodpis. Google waypoints: https://goo.gl/maps/XtJUi
Also there are a bunch of towns off this route. Look at pictures of Kovachevitsa, Gela, Mogilitsa, Arda, Smiliyan or Shiroka Laka, see what looks nice to you. It's hard to tell you exactly what to do as there are so many caves, hikes, and random things to see. You would be better off not planning at all and having B&B folks tell you what to do.
Is it worth it to consider the Black Sea area, or it is more of a for
beach holiday instead of culture and travel? We live on the ocean.
I just went back to the seaside (Nessebar) for the first time in probably 15 years. If you are planning on going in Summer, I would probably just pass.
Second phase is to check out Lake Ohrid
and Skopje, North Macedonia.
Pass or one night.
With a possible day trip to Pristina, Kosovo
Also pass. Kosovo is pretty cool though, I just went on a small roadtrip a few weeks back. Highlights were Gračanica Monestary, Prizren (very cool town, reminds me a lot of mostar), Monastery Visoki Dečani and the Patriarchate of Peć.
and possibly driving through Serbia to fly out of Sofia,
You can't do that. If you enter through Kosovo you cannot exit via Serbia. Even if you get past the internal soft border which you likely won't they will stop you on exit as you don't have a Serbia entry stamp.
One of my main thoughts is where are you going to find a car rental company that allows all this, especially Kosovo.
And my biggest question and concern, how are the roads, driving
conditions, and border crossings?
Roads are fine everywhere you plan to drive. Driving conditions are fine as long you pay attention and don't drive slow in the fast lane. Border crossings are non events, most of the traffic in the Balkans is N/S (Turks driving back and forth to Germany, TIR traffic to Europe, Bulgarians driving to greece etc), not E/W so the borders are pretty fast.