It looks like it might have gotten burnt foliage from a heat wave, from what I can see in the photos. The newest foliage that is more vertical seems entirely unaffected, and the flatter leaves at the tips seem to be the ones that got crispy. I'd suggest mulching around the base of the plant to retain moisture, trim off the burnt parts to better track if this is getting worse or was a one time event, and when you water, it is probably better to water for at least 5 to 10 minutes with spray irrigation and not do this every day unless necessary to get new plants established. Cordyline australis cultivars should actually be pretty drought tolerant and get by with just once a week deep soaks even in 100 degree weather, if the soil is well mulched. On the other hand, this is a species that comes from New Zealand where it seldom gets to be 100F for weeks at a time, so it could just be the heat. If this continues to happen every summer, even as the plant becomes deeper rooted, this would seem to indicate that in your climate with your summer heat, it would prefer some dappled shade at the hottest time of the day, rather than full sun. The green leafed Cordyline australis species is certainly much hardier/resistant to sunburning, the newer colored forms are more prone to sun damage with high heat.

I live in the UK and see Cordyline australis everywhere. Indivisa isn't so common though, I know there are some in very wet parts of the UK like some gardens in Northern Ireland and Devon. Occasionally we have a cold winter and there's a mass die off inland, but it's no big deal really. Most come back very strongly from the roots, especially if they were a large, mature plant before they died in the winter they reach their former size in a couple of years and come back very fast from a strong root system with fresh, new growth. What actually kills them is when you see well-meaning but misguided people wrapping them up in winter with bubble wrap and frost fleece. It's the worst thing to do in my opinion, these things hate to be soaking wet in winter and the frost protection people put round them just makes them rot. I don't protect my ones that are in the ground. If they die back they dieback, the soil doesn't freeze deep enough in Britain to kill them in most cases. Container ones i just put in a cold greenhouse to keep them from dying from the damp of winter which is the real cause of death. I have noticed a lot of variation in the plants grown here. Some are very narrow leaved, but mature plants are usually wide leaved, not sure if the leaves get wider over time or different varieties are tougher. I suspect the narrow leaved plants are the "Atlantic green" variety whereas the wider leaved ones are generic australis with maybe some indivisa introgression, although according to something i read somewhere the narrow leaved varieties are meant to be hardier. I've also never seen a mature, large plant of the red, pink or variegated varieties though i know some examples exist, they're probably less hardy and were perhaps selected from stock from the North island.

Never tried to divide these but the shoots contribute to a very ornamental presentation of the plant. Cordylines are often planted in clumps as they can look very attractive this way. If it was me I would be nipping back the main trunk to encourage the clumping look. I have grown them from stem cuttings quite successfully. If the shoots have a reasonable amount of roots attached you may be successful.

The blue is not fungus, its residue from fungicide thats been used before you bought the plant. Many fungicides leave a blue or white powdery residue. Nothing to worry about other than that it's toxic if it gets ingested, licked by cats.... In production they keep them very humid and will use fungicides to keep the leaves in pristine shape for sale as the tips often go brown. They should use a mild detergent to clean them for sale. Cordylines also often shed lower leaves when placed in a new environment especially a much drier one. This is also not something to worry about, though you could improve humidity one way or the other. Look at the roots if they are nice and white thus healthy it's not a root problem. Root or stem rot will cause the lower leaves to drop off very quickly. It's unlikely this is your problem indoors though. Older leaves will look spotty and brown parched, before they abort, also no problem. However the large "water mark" in the healthy pink leaf is symptomatic of something, probably dry air in your case or possibly over watering, no drainage at the roots, or even lack of water at the roots, check all these things. It could also be burnt from being too close to a window or a lamp. To increase humidity sit the pot on some pebbles in water don't over water instead. They do like a moist soil but not a boggy one. Indoors they also want the brightest light possible short of actual sun, certainly in summer as the leaves can scorch through glass. The leaves are thin in most cases and burn easily or dry out easily especially the new leaves. Also don't feed more than just a tiny amount once a year indoors. Cordylines quite frankly don't make the best house or office plants because they love humidity and warmth so much and because the leaves are most often quite thin. To recreate ideal conditions would make anyone using the room uncomfortably steamy. However if your climate is good you can place them outside for a few months (even just summer) then inside for one, rotating them as you go, or just inside for winter. Other wise look for the varieties with the thickest leaves to try, there are one or two that with some help should do better as a result. Off hand C. fruticosa "Vanuatu" is one, shiny largish jet black leaves, arranged beautifully. I've never had any problems with it's leaves outdoors in full sun and drenching rain, humidity and a dry warm winter. Some of the rounder paddle shaped small ones or the larger leaved reds with the same thick leathery leaves might also do. Dracaena are all round are far better than Cordyline fruticosa for permanent indoor use. You are unlikely anyway to get a fungus problem on the foliage indoors only possibly at the roots or the cane. Fungicide could help with that but then it must be mushy wet and black to be that. Lower older leaves don't always look their best indoors but it is technically better to wait for them to drop off rather than cutting them off or the plants stem (cane) might get thinner and thinner in the process, bending over and ultimately die. Hope they've survived meantime anyway!