Is zakat obligatory on a plot of land that I intended for trade and sold in installments, and on another plot of land that is offered for sale for the purpose of trade or building upon it and selling it? And is it obligatory on an apartment that I bought in installments for dwelling, trade, or rent?

If you purchased land with the intention of trading it, then it is considered a trade asset and zakat is due on its value every time a hawl (lunar year) passes over the original capital. Its stagnation does not negate the zakat. Stagnant trade assets are subject to zakat for one year when they are sold. If you intend to sell the land after developing or building on it, this does not negate the zakat; it remains a trade asset that is valued and on which one-quarter of one-tenth (rubu' al-'ushr) is paid annually, or for one year if it was stagnant. As for the apartment you purchased, and you are undecided whether to sell it or rent it out, there is no zakat on it due to the intention of trade not being definitive.