I wish to conclude one thing about this that whether mining solo or having a mining pool where others work for us with their computational power, what exact type of control over transactions do we have?
The protocol requires that you ONLY include VALID transactions in your blocks.
As long as the transactions are valid, you can use any criteria you like to decide which transactions you include.
Since most miners are interested in profit, most of the existing mining software and mining pool software uses a default setting of choosing the transactions that pay the highest fee per block weight. However, some of the software is configurable, or will allow you to override the settings with specific transactions. Also, you can always create or modify the software on your own (or hire someone to do it for you).
I mean do we have the control to decide the fee/byte we want in order to be included in the block that we may (probably) find?
Yes, the rules of the protocol allow you to choose any valid transaction using any reason you like.
If yes, is it like a slab that we need to choose to include the given sats/byte transactions included only?
I don't understand this question.