Think of it like 1) taller joists, or 2) the second level of a house (without the added load of a second floor). There's no reason why the load from the upper level can't be transferred through the lower level to the footings.
As to size, the upper deck can be as large as the lower deck, and if there's no cantilever on the lower deck the upper can extend to whatever is allowed in local code for cantilever.
Regarding your updated questions...
Yes, that's acceptable, assuming the beams will carry the additional lumber load safely.
No, that would be far more expensive and time-consuming that your other options, the best of which (from what information I have) would be to simply raise the existing deck.
What you're trying to do appears possible, with a few extra parts. See here (forgive the crappy art, I made several edits):
The two red rectangles are the beam-post braces I would add (minimally) to handle the horizontal loads which can no longer be handled by your odd post.
The post I drew where there isn't one is just a short floating section used to fix the two beams together.
What you need to build so your post does not need to be buried in concrete is a cantilever. Extend the 45 degree horizontal beam some distance inside the pergola. The longer you make it, the less load you will have where you join it thanks to leverage. I would aim for as much inside as there is outside. Fix the end to another 45 degree beam (shown in red). This needs to be rather strong as it will be handling roughly the weight of that corner of the pergola. Except that end will be trying to move down, not up. Think of that joint as hanging a large weight on the red beam.
I've shown the red beam as resting on the 2x6 beam, but attached to the side of the 2x8 beam. It should be easy to find some kind of metal L you an screw to the 2x8 and rest the beam onto.
The connection between the two 45 degree beams could be a large bolt down the center, with appropriately large washer. They will be about 6 inches apart vertically as the red one is over the 2x6 and black one under, so that would likely be easiest. But it's also possible to insert a short section of wood and use steel plates to fix both ends to the beams properly.
I also suggest resting the 45 degree beam on top of the post, with some minimal fixing to keep it from moving around. I would drill a hole from the top and drop a rod which will not rust in it. Same deal where the 45 degree beam meets the tiny floating post. A lag bolt from below would do fine. Or any kind of screw you can get that's long enough, it won't be pulling anything but its own weight.
With that setup, your odd post does not even need a brace to the horizontal beam to hold everything up. But it will help overall stiffness to have one.
If you go the "10 foot post into concrete" way, I suspect you will find your post slowly tilting towards the pergola over the years. But that's just an educated guess.
Best Answer
The problem with the pipe idea is that unless you're going to run it at an angle, you'd never be able to get a pipe through several consecutive rafters.
You may actually be OK with lag screws - a properly installed 3/8 lag screw should provide an enormous amount of withdrawal resistance. Here's a sample chart from Masco:1
I wouldn't be horribly concerned about the roof members, as long as you are attaching where they have full thickness and minimize the distance of the cantilever.
Your most likely points of failure are going to be the chains (see this answer), or the hardware on the swing itself. In general, working with rated fasteners installed they way they were intended to be used is a better idea than "winging it".
Note that I have no idea of the condition or construction quality of where you intend to hang your swing to, and I am not a structural engineer.
1 Not to be taken as a product endorsement or as representative of other similar products. You will always want to check the ratings of the hardware that you actually intend to install.