Your receptacle tester definitely should work normally. Nothing about a 3-way makes it weird, unless you mess up, which is very easy to do with 3-ways.
Why don't your receptacles work? Red black and white spaghetti - who can tell anything in all that schtuff?
They did the minimum legally required: they taped the white wires to indicate not neutral. Why on earth stop there? Let's get some blue and orange tape, and tape-wrap those wires for what they actually do. And let's see if it doesn't make a lot more sense.
- Pretty straightforward. Any wires still black are definitely always-hot.
- Any wire still white is neutral.
- Blues are messengers - they both do exactly the same thing, and you're free to swap them anywhere along here, so, no need to distinguish one from the other.
- Orange is switched hot intended for the bulbs (or you say receptacles? hmm.)
My guess is, you got the black red white spaghetti all mixed up and grabbed the wrong wire. Label the wires properly and I bet it falls into place.
By the way, using two cables like that, C3 and C4, is illegal because you have imbalanced currents on the wires. The power goes out on C3, but comes back on C4. This will induce eddy current heating on any part, box entry, staple or nail between them. The right thing is to use /4 or /2/2 cable, which gives you 4 in one cable. ----- Or run /3 and /2: the /2 would connect the two lamps to each other, connecting switched-hot and neutral. C3 would be both messengers and switched hot - out on the messengers, back on the switched-hot, balanced. C4 would be simply a "spur line" to the other lamp, out on orange back on white, again balanced. C4 could even be a "tee" to another location.