What can I do about sunken pavers that flood when it rains?

I noticed the same: the patio itself has more than 6 different slopes and the photo #1 reveals where the water mostly sits.

Because it doesn't sit in the center, nor on the entire all the suggestions of stones beneath, pipes beneath, etc, are all in vain. The patio is not the source of the problem. This needs to revamp the entire form of the region around it so that the water is not flowing from the landscape onto the patio. A Step up Patio by reducing all the regions around it by 3" often can make a world of difference. As far as the unevenness of the blocks, that explains by the weeds and grass have different sized roots and will force those out of place. Those weeds and grass need to die before that project is finished. Vinegar+RockSalt may not help more than lift up each block and dig out the weeds with a pitchfork.

As firstly you look at that picture, what is missing?

A curb all around the patio.

When that is the case, that you gotta lift all the pavers, you may as well consider:

Do we really like that patio in that trouble area, or Do we prefer it somewhere else, with less trouble, much better?

Although many resolves exist, [such as: 200lbs of Rock Salt beneath those pavers after you place a 5 inch Curb], facts are:

When all nontemporary resolves entail that ya gotta move a Patio, you may as well just move the patio, plant a tree where the patio was, start over, and consider it Done, by doing the work, getting it right once, to last for decades, rather than do 1/2 that work every 5 years til ya die.

Getting it right? Always involves alot of work to reduce the long term work.

Resolve to plant a Tree where the wetness is to make good use of the moisture, After you concentrate on excavating where you actually want that patio.

This involves Mixed Stone and Sand and Salt as the base Layer placed at the Clay Layer [you may optionally heat with a flamethrower, or a Bic Lighter and Lemon Pledge, until that clay is like bricks], and you place that stone over the Sand that secures 4 Dispersion Pipes connect like an X. You add your last phase of Stone over that, to place your 4 Corner Patio Blocks in that Area that will be Free of Water and Weeds, for as long as you maintain that Clean and with a very high PH That nothing green survives.

After that, you frame out those 4 corners to pour a Curb.

Shape of that curb matters not, keeping thise 4 corners cornered is what matters.

Once that curb Sets, you can place your other blocks, side by side, be that inside a cement oval curb or inside 4 Circles as Curbs or inside a large Sexagon, or even Checkerboarded with a Smaller 36 Block Solid Color in the Center, it all looks so much better than:

Wet Bland Square Patio in a quite hopeless condition of severely uneven greenness, all caused by the roots of undergrowth and weeds, which caused by 'the lack of a curb allows grass clippings, spores, dandelion seeds, moss etc to get in, die, rot and become soil for the next seeds', which all began 7 to 15 years ago, when the quite Bland Square Patio initially misplaced...