I have a very relaxed attitude to 'weeds' compared to many, and what I'd do is quite different than some...
I have cold compost and throw all sorts of things in there that people warn about.
I don't have issues with weedy compost, but I mulch heavily nearly everywhere.
I wouldn't put seeding perennial weeds in the compost though.
Do you have a stoloniferous lawn grass? Pardon the word, but trying to describe it takes ages! If not, how about putting the sod at the bottom of your garden, upside down?
Even the poorest soils are usually reasonable at the surface. The sod, being mostly mineral soil, won't help composting, but I'm all for using paper/card. Rip it up a bit and throw it in. Make sure the scraps/card are mixed up a bit.
The best compost is made with a wide variety of ingredients. Mine is draped with old cotton gardening clothes right now, waiting for the next layer of scraps.

I thought the heat from the composting would kill the seeds.

There are a few issues here to think about.

If the lawn clippings and weeds aren't going to seed, the odds of them spreading are fairly low. Weed seeds are what you want to avoid in the compost, for the most part, but the non-propagative parts of the plants will only add nutrients to the mix.

If you are building a large enough pile, with the proper ratio of ingredients, it will create enough heat to destroy most weeds' viability. If the pile is too small, it won't heat up enough, and weed seeds and grass roots will survive even a long composting process.

Sod is generally a good thing to add to a compost pile because it introduces microorganisms that will help in the decomposition process, but the risk of those dreaded stoloniferous grasses remains. the way to deal with that is to drown them for a few weeks first, then add them to the pile (they'll smell for a day or two).

Everything you've mentioned except the cardboard is considered a "green" material. You will want the ratio of "browns" to "greens" to be about 3/1, which will mean shredding a lot of boxes. If you can locate some old leaves, hay, or wood shavings, it will help to create a more actively decomposing pile.

Since "weeds" are simply plants you do not want growing where you do not want them growing and those "weeds" take nutrienst from your soil recycling them on your property in some way simply makes sense. There are those that srike fear in others by warning them that any "weed" put in your compost will result in you having tons more when you use that compost and that is simly not true.
As long as you use care to be sure that the "weed" seeds and roots are not included when you put those "weeds" in your compost pile most all will not be any more of a problem than anything else. Some people also seem to think that grass clippings can root, like some plant stems will, because they see grasses growing into beds to which compost has been added. If "weed" seeds or root bits are put into compost, and that compost does not get hot enough to kill them, or the seeds are not buried in the compost long enough, they may well root. However, most often what I have seen is that the grass has sent roots into that nice, nutritious soil or seeds from outside the garden has been deposited there. The source could be birds or maybe even the wind.

I do the same as feijoas. The only thing I won't put in fresh is bindweed. And that goes in after it has been hung up on a bush or somewhere to dehydrate. My compost never heats up. Any weeds that do germinate after the compost is spread are easy to pull and go back into the next batch. Weeds grow on my land and therefore any nutrients they absorb go back onto the same land. To throw them away would be to rob the garden.

I don't know how much sod you have, but if it's more than about 10% of the volume of the compost pile, it can be a detriment. All that soil doesn't really add anything to the compost, but its weight will compact the pile and reduce air infiltration, which you need to get compost. I tend to turn it upside down on the bed or else plant it in a bare or eroding spot rather than putting into the compost.