Ways to find out amount of tax before selling?

Does the amount of tax I have to play subject to any other side effects that are not usually mentioned in the description of this? For example, does it matter whether I have other (salary) income in the year I'm selling a sizable amount of stocks - would it be wise to sell the stocks in a year where I have less income (maybe due to age/exiting the workforce)? Or do these types of income simply not interact with each other? (I do not mean schemes like moving stocks around between banksover the years to somehow optimize the "sell buckets" or things like this.)

It depends. If you exceed the 1000 € on gains, you are subject to the mentionned tax.

But you can apply for "Günstigerprüfung" so that the tax office checks whether it is more favourable for you to pay this tax or to have it applied as regular income. Especially if you are in lower income regions, this might save the one or other euro.

The actual tax I'll pay depends on the gains I made, which in turn depend on the "buckets" in which the stocks are sold (assuming that I rebought them regularly over the years). But the data for this (when, how much, and at which cost I bought something) is with my bank. Are there regulations about them having to inform me about those buckets? I.e. say, for arguments sake, assume there is a large amount of stock X, bought with many smaller transactions over decades; are they obligated to give me detailed info about which buckets (and hence what amount of gains, and hence which amount of tax) would occur if I were to sell, say, a quarter of X? Or is it on me to keep track of the individual transactions and do a "shadow calculation" for this?

I don't know of any such obligation. Some might do, some don't. It is better if you keep your own records in order to determine how much to sell.