The quickest method of getting rid of those "weeds" would be to till them in, option 2 in your post. Some may tell you, "Oh, do not till those weds in because that will just encourage more to grow." However, unwanted plants will grow there anyway since Ma Nature abhors spaces with nothing growing there. Seeds from plants come in on the wind, are planted by birds, and are dropped by animals as they travel through the area.

Option 3 would also work but take a longer time, 4 to 6 weeks, but would involve less work, but would interfere with laying sod this year. This is more environmentally sensitive than tilling would be, however.

Option 4 is not a good choice for environmental reason and could take the same 4 to 6 weeks between applying the herbicide and laying sod since you would need to wait for what ever product used to dissipate.

What floral said - there are some weeds that herbicide won't affect, and digging is the only way to go. The smothering technique, of which I am very fond, only works well on annual weeds. Let us help you identify what you're dealing with, as floral said, and plan your attack from there.

Another thought - I don't like herbicide use either, but sometimes that is the only way to go, but one must go very very carefully. So it's your classic three-pronged attack.

I'd go with hand pulling or manual removal as well. It's not a large area and should be quite manageable and hand pulling gives you the option of being very thorough with the removal of roots or any rocks, etc. And there is something very zen-like and satisfying about clearing an area of weeds manually. Loosening the soil while doing this will help to prepare the soil bed for sod as well.

The second option would be to use a herbicide. Also very fast and effective, provided there is nothing else around in the immediate area that could be harmed. And you may need more than one application. Most herbicides, like glyphosate for instance (RoundUp), breakdown very rapidly once in contact with the soil and you can replant almost immediately, within a week to 10 days.

The plants growing there now are utilizing nutrients in the soil to grow and removing them will also remove those nutrients while tilling those plant in will recycle those nutrients back into the soil. Tilling would also allow pretty rapid sodding, probably sooner than any other method. Tilling what is growing there now would also add some organic material to the soil that may be needed.

Keep in mind that glyphosate products have been listed as potential carcinogens by the World Health Organization (WHO) and that California has said manufacturers must put that on the product labels.

But mechanical tilling will not deal with persistent perennials. Annuals can be successfully turned under but that method just won't work for all weeds. Hand digging will deal with perennial roots. That's why identification would be useful. And I'm sure no one is advocating throwing away pulled weeds. They can be composted and returned to the soil.

Looks like less than a whole days' worth of work, pulling, which I do all the time. But working that hard to install expensive-to-maintain, work-intensive grass instead of... anything else is not a concept to which I can relate, especially if you are renting. Whatever all of that is, you could just mow it & call it lawn.

If you do try a smother, do not put soil on top of newspaper. Use mulch &/or other organic material on top of strong, corrugated cardboard. Newspaper is not strong enough to thwart the growth of many plants, and putting soil on top will likely result in a whole new crop of some kind of sprouts... that would then need to be pulled or otherwise dispatched.