Leaf splitting is caused by the plant not getting enough water. This could be from not being watered often enough, or from having dead roots which are unable to supply water. But it takes more than a week for leaves to split, so the problem began long before you bought it.
It's hard to say what the white spots might be without a photo. They could be mealy bugs, scale insects, spider mite webs or mineral deposits.
What are the temperatures inside your home?
I would suggest repotting the plant unless you know it was repotted recently. Phal. roots must have lots of air around them or they suffocate. Old medium decomposes into small pieces and dust, which plug up the air spaces.
The medium it is in now is a type often used for Phals. Get your materials together first: the potting medium, and the pot. The pot should just hold the roots; it is best to use a smallish pot for a sick Phal. It is possible the current pot will be the right size. Clear pots are helpful because you can see the roots and tell whether they are green (moist) or silvery (dry.)
Then unpot the plant and look at the roots. Young healthy Phal roots are white with green tips. As they age they are silvery when dry, and green when wet. Even older healthy roots may be light brown when dry.
Don't remove roots unless they fall off on their own. You can't always tell which are dead. It is possible for a portion of the root to look dead, but portions farther towards the tip might be fine.
Repot into fresh medium so the bottom leaf is just above the medium in the pot. Water thoroughly, perhaps by soaking in a bowl of water in the sink for a few hours. When done note the weight of the pot after soaking. As it sits in your house it will become lighter. Water before it becomes completely dry.
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