I think doing really solid engineering calculations on this, which is always the right thing to do, would be pretty difficult. It's easy enough when you just have a span between two poles but when that span is supporting a number of other span at angles to another span ... ugh.
I might just try to take a very conservative common sense approach and see if it works, erring towards caution. Your mileage may - just suggestions here.
I don't really know how much tension you can put on a 13' 4x4 steel post - it would depend upon the gauge of the metal etc. With a 3' x 18" base in the ground ... I don't see it being an issue for this fairly lightweight application.
Before I go any further - strongly recommend you use low voltage 12V lights - much safer and simpler to install in a code-compliant manner. The electrical code requires a messenger wire for spans over 40' and requires strain insulators for the attachment of the messenger wires. Although with GFCI, 120V is much safer than it is without, I'd stick to low voltage.
Also keep in mind that wires, chains, ropes, etc. under tension can be dangerous if they snap, or come loose - the tighter they are, the greater the tension, the more energy they store, and the more dangerous if they come loose.
I am going to call the wires in green in your drawing "messengers" instead of "guy wires." I usually call guy wires wires that go from the post to the ground to help keep the post straight. I'd really try to avoid using guy wires, people are always tripping on them or walking into them. You want posts that are strong enough without them.
The main thing you have to look at here is tension in the messengers - that's what is pulling on the posts. The tension on the messengers depends on how tight you stretch them, and the weight they're supporting. If you want a really laser straight messenger, that will generate a lot of tension, I would NOT attempt that, I'd allow quite a bit of sag.
The string lights are also under tension. In fact, I am skeptical you'll be able to find string lights that span over 50' unsupported. I believe you'll need to string messengers between the messengers to support the string lights.
You'll want to use wire for the messengers that's sufficiently strong, but I wouldn't go overboard, since it's just more weight to support.
I think I'd string the pole to pole messengers with a foot or so of sag, then string the messenger-to-messenger messengers with a couple feet of sag. You could attach the lights with zip ties or maybe wrap the light wire around the messenger as you go, and tape it at the ends. See if the tension is something you can tug on and make it move a little with one hand at the poles. You know that if you can move it a little with one hand, it's nothing that's going to make your posts lean.