What temperatures and humidity are you maintaining? Calathea like it humid. Dry indoor air -- especially common for those of us in the northern regions over the winter -- is stressful to them. They like it rather warm -- temps in the 70sF or up. Keep the plant away from cool drafts or from furnace/heater vents. Having chilly air or hot dry air blowing on it is not a happy situation. Watering can be an issue. Depending upon where you live, watering may need to be adjusted throughout the year. If you keep your home cooler in winter the allow the plant to get a bit dry (not parched, not bone dry) between waterings. When temps are cooler, the plant slows down (so to speak) and does not need nor make use of as much water. Keeping it too wet at such a point, can result in root rot or even stem death due to bacteria or fungi. In the heat of summer, they can use more water and so can be kept damper. As other have said, avoid direct sunlight -- bright indirect should be good. Keep in mind too, that plants require time to adjust to new conditions. How much time and whether adjusting is even realistically possible depends on how radically different your conditions are from the original growers (not the BBS you may have purchased it from -- the REAL growers ... as in the nursery that supplies the BBS) as well as the type of plant itself (some plants are simply far more delicate than others). Constantly switching conditions as you seem to be doing -- while completely understandable -- can keep the plant in a constant state of stress. Do not fertilize any more for a while! A stressed plant has enough issues to deal with. Fertilizer is not a "magic cure-all". It can actually make the situation worse. What is the plant potted in? The media in which most BBS plants are potted is usually -- to put it mildly -- absolute "crap." Typically the media is heavily or entirely peat based which means that it holds a great deal of water for a long time BUT once it dries out it tends to become hydrophobic (water repellant) meaning that it takes "forever" to remoisten. It is possible to have the outer ends/surfaces moist while the center of mass is completely dry.

Sorry for the loss, consider it part of the learning curve in gardening. For the future, best to keep in mind (I learned this the hard way too). Best not to buy living plants from florists, their knowledge tends to be more about cut flowers than living plants. My example: I bought a tropical plant (which I later learned goes dormant), but the florist who sold it to me didn't know nor did they tell me. So when it went dormant, I threw it out thinking it was dead. Bummed me out to learn, not dead, it's just dormant for the season. $15 down the drain. Never again!

Yep, it's definitely dryness response... I had forgotten to water mine, and that's what happened. Causes can be being rootbound, drop in humidity, and change in environmental conditions. Its certainly not scale, mealies or spider mite. As always, look at it with a jewellers loupe, or magnifying glass... As a keen gardener, I always carry a loupe when doing my indoor/outdoor rounds. This particular phenomenon is a sugary excretion - if not noticed quickly, it dries into a sticky crystalline spot, and leaves a transparent spot on the leaf underside. I have cleaned it off with soapy water, and a fresh water wipe afterwards Any little bugs will have been attracted to these sweet secretions, they are not the cause.

You could try, but calathea is normally propagated by division. You see, its stem is underground and is called rhizome, quite like peace lilies, and the leaf goes out direct from it, with a petiole conecting both [sorry if theres any mistakes I dont remember quite well plant morphology]. Because of this, leaf propagations are very hard, normally you would divide the plant by unpotting it, if its big enough for the dividing, and cutting a bit of the rhizome with leaf and root. But you could try leaf propagating, just quite rare.