How does strike price and assignments work

Most equity options are American style (just a term - it has nothing to do with nationality) meaning that they can be exercised anytime. Also note that what the stock has done in the past is irrelevant for these options. If the stock was over 50 in the past but is at 45 now, it would not get exercised. Why would someone opt into buying a stock for 50 when they could buy it for 45 on the open market?

Options that are "in-the-money" (meaning the stock is above the strike for a call and below for a put) are typically "auto-exercised" by the broker at the close of the expiry date to avoid any timing problems.

Also note that it's almost always better to sell an option rather then exercising it. If you hold an option with a strike of 50 and the stock is at 55, the option is almost always worth more than 5 for reasons that may be beyond the scope of your question (and can be researched here and lots of other places).

I feel like you're thinking of these as "knock-in" options, meaning that if the stock reaches the price at any time then it can be exercised. While such options do exist, that's not how typical equity options work. Whether the stock "reached" the strike price in the past is irreverent. What matters is the price when the option is exercised. They can either be exercised at any time regardless of what it's done in the past, or they can only be exercised at the end of the expiry date.

If the stock rises to $45, it has reached the strike price, regardless of the time of day that it hits that price.

American style options can be exercised at any time (equity options). According to the CBOE, only about 7% of them are exercised before expiration . There are several reasons for this:

• Someone just wants the position in the underlying

• The option is deep in-the-money and it trades at a discount. This presents the opportunity for a discount arbitrage.

• There is a pending dividend which also presents an arbitrage opportunity.

Options that expire in-the-money will almost always be exercised since it's OCC policy to exercise them unless they receive 'Do Not Exercise" instructions from the option owner. This is called Exercise By Exception.

At the beginning of the week, a stock is $45. A one week $50 call is purchased. On Wednesday the stock fluctuates between $44 and $55, closing at $49.

What price info is used to calculate if an option reached the strike price?

The underlying reached and exceeded the strike at all continuous times it traded at or above $50. That doesn't necessitate that the buyer of a call in that option series is obligated or interested in exercising it.

Is it any price from day of purchase to time of expiration?

Yes

Or is the closing price of each day taken into consideration irrespective of what prices a stock reached through out the day?

Closing price (daily) isn't more or less important than the price at any other time, except at expiration. Generally 4:30pm on Friday.

Assuming the strike price is reached during any of the days preceding expiration, can the option be executed and assigned or it will get assigned only after Friday's expiration?

Like others have said, the option can be exercised at any point, regardless of price. The price could be below the strike K_c and still the option holder could exercise their right. Exercising OTM and exercising early are rare, though.