Is it permissible to deal with banks for taking and investing money, while paying installments, and depositing a portion of the money with them to cover calculated interest, then transferring the money out of the banks after settling all dues? And how is Zakat calculated on these unstable balances, both credit and debit, with the bank, in addition to income and salary?

It is not permissible to invest in interest-based banks. What is gained from such investment is usury (riba) that must be disposed of by spending it on charitable causes. It is not permissible to repay interest-based loans with it. Zakat is obligatory on the principal amount of money deposited for investment if it reaches the nisab (minimum threshold) and a hawl (lunar year) passes over it, as well as on the profits if the banks are not interest-based. When zakat becomes due, debts you owe are deducted from your zakatable wealth if you do not have other surplus wealth sufficient to cover those debts. Debts owed to you by others are subject to zakat if the debtor is wealthy and capable of repayment; otherwise, zakat is paid on them for one year upon their receipt. Zakat is obligatory on savings from salary if they reach the nisab and a hawl passes over them.