I am building a 28L x 8D deck with a metal roof off the house. The deck is going to be only 12 inches off the ground. I’m using 2x10-8 pressure treated for floor joists 16 O.C. And Trex.
With my 28’ span I want to only use three posts--one on each end and one in the middle. So now that makes my span 14’, with putting 2x10 on each side of post which will be using carriage bolts and hangers.
Will that be good for the span for the deck? Can I do the same for the roof?
For the deck, I doubt that any inspector would accept 2x10 for a span of 14'. That's uncommonly long for a residential deck. You'd need 2x12, and I'm not even sure that would fly. You really should have two intermediate posts. I'm reasonably sure double 2x10 would span the 9' or so that gives. Otherwise plan for triple 2x12 to span 14'. Ask your local inspection office to be sure.
The roof probably has lighter load requirements. I'd expect that double 2x10 would work, but it depends on local snow load. Again, ask your local inspection office (or sweet talk a lumber yard into helping do the math).
Also note that any structure attached to your home must have full frost footings in most jurisdictions. That means poured or otherwise approved concrete bases to your local depth requirement, plus posts approved for underground.
And as SteveSh says, don't rely on surface-mounted beam members. They must bear on wood at least 1" deep. Notch them into your posts. Since you have a flush beam they need to be tight together and connected, so you'll need at least a 2½" notch at the outside of the posts. (Your drawing has them spread, which won't do.) Use size-appropriate joist hangers. It's convenient and precise to set one beam member, face nail the joists in place (flush on top), then install the other beam member and add the hangers after.
FYI, what you've labeled "lathe" are actually called "purlins". (And it would be "lath". A lathe is what you use to spin wood when making baseball bats.)