You might be able to use a fish tape to pull the new wires. About $10-20 for a 25-50' long fish tape in most hardware or electrical stores. This is a rectangular stiff steel wire, in a coil, with a hook on the end. You hook & twist the existing wire to the fish tape, tape over the twists so it is smooth, and then use conduit grease (honest) on the fish tape to let it slide easier. As you pull from the far end of the old wire, it pulls the fish tape through--as best it can be done. You may need to access the wire and help it along, or the old wire may be pinched or epoxied in place in some spot. But, if you can pull the fish tape through, then you twist on the new wire, and reel the fish tape back out, pulling the new wire through where it was.
As to tees and side runs...I'd just pull the main wires through, and make the tee connections afterwards. Pull out some slack, make the connection, push the slack back in and leave it there for future access.
How you make the connections (crimps, solder, etc.) is the subject of many discusssions and the bottom line is probably if you do the connections properly--you can do them however you'd like. There's good and bad to both soldering and crimping.
If you can't run the fish tape through, it is time to look for a new place to run wires.