The pay period is from Monday 12/23/19 till Friday 01/03/20 and this pay period gets cashed into me on 01/10/20 (Not sure if this piece of info is relevant). Do I have to include this on the 2019 tax year or the 2020 tax year, or do I pro-rate it by days? Or, can I chose any of these options?
Income is considered to be part of the tax year in which you receive the paycheck. Thus, even though you earned it in 2019, it'll be on your 2020 W-2.
This is fair, because you'll have a similar situation at the end of 2020. IOW, the "extra" 2019 income on your 2020 W2 will be offset by the 2020 income that will be on your 2021 W2.
You're over-thinking this. Assuming you're an employee, you use whatever amount is shown on your W2 form. If you're an independent contractor, whatever's shown on your 1099s, likewise if it's income from stocks &c.
The default accounting rule for personal income is "cash method". Under this method, the tax year for income is not based on when the income is earned, but when you have "effective receipt" of the money, which as MTA says in the comments, is when "you have access to the funds 'without substantial limitation or restriction'." If you have an in-office mailbox, for instance, and your paycheck is placed in it on Dec 15, but you don't bother picking it up until Jan 1, then your effective receipt is Dec 15: even though you didn't actually have possession of it until Jan 1, picking up the check and depositing it is not considered a "substantial limitation". On the other hand, if your company is late on payroll and doesn't get the check out until Jan 1, that will probably be the date that will be considered your "effective receipt". Basically, as soon as you either have the money, or could have the money if you wanted, you have effective receipt.