Can I solder the power source directly to a rectifier to make it DC?
Yes, you can solder components directly together. There's no law of physics that says you have to have a circuit board at all.
If you do it, make sure all your wires can't touch each other when they're not supposed to. That would be a short circuit.
I plan on using a protoboard using the DC power but I need the board to be able to handle 2Amps and from what I found regular breadboards can't be used so I've been looking at veroboards.
What are trace lines on veroboards?
This question has already been asked - the latest answer on this question estimates you should be able to run at least 5 amps through a veroboard trace.
Lastly would it be safe if I solder circuit together directly instead of using a protoboard?
The main thing to worry about, that you don't worry about with a protoboard or breadboard, is accidental short circuits. After building the circuit you'll find out how rigid it is. You could use something like electrical tape, heatshrink or even hot glue to keep wires separated.
The electrocution risk would be the same as it would be with a protoboard or breadboard. The usual advice applies: don't touch the high voltage bits, or better yet, if you aren't confident then let someone else do the high voltage bits.