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Rational means the same thing it means in other contexts. If a rational man is faced with buying the same product for two prices he will chose the lower as a rational choice.

How often do you see "the same product [sold] for two prices"?

If I chose to buy from a local grocery because I like family owned stores do I act irrational?

If I buy more expensive product because I do not have time to look, do I act irrational?

If I drive around and waste time and money to buy something useless but what I like, do I act irrational?

If I am motivated by the emotions or impulses and not by calculation, , do I act irrational?

If I give money to a bum on a street do I act irrational?

If I chose to work less in order to have more free time do I act irrational?

If I buy expensive caviar instead of cheap gruel do I act irrational?

If I decide to save money and then die leaving them to the inheritors, do I act irrational?

Is my best interest different from the interests of my close relatives or from my friends, or from my country? Which should I chose?

Is my best interest to enjoy money now or to save them to get better return later?

If I spend money on myself instead on my family, do I act irrational?

If I can steal without being caught or punished, do I act irrational?

If I vote for the government which will serve my private interests by handouts, do I act irrational?

Let me take a stab at answering your questions:

Every day.
No.
No.
No.
Maybe.
No.
No.
Not unless you're starving.
No -- you're helping to ensure your lineage survives.
Economics can't answer that -- your best interests would consider all those things.
Depends on how much you enjoy having things now, compared to future security.
Maybe.
Not if it doesn't bother you -- this is why we need police and jails.
No -- that's why it's important to show people that the handouts hurt everyone eventually.

Attempts to complicate the questions and alternatives are not refutations of the definition of "rational" and are easily handled. I gave a example with two alternatives as a basic concept. Extending it to multiple choices and even multiple choices over time does not change anything.

Since most of your complications are easily handled I will only address a couple.

Of course, I can see the same product with two different prices that is not unusual. I have even seen them side by side with two different prices.

Inclusion of a preference for "local" stores in a decision-making function is not anti-rational merely an extension of the criteria used to make that decision. You have added something not considered by another who doesn't care about it.

Search costs are also part of the decision-making process and do not change matters wrt rationality either.

Purchases are not required to be "useful" to be rational. If the economy was restricted to "useful" purchases it would collapse across the globe.

Emotional decisions are not necessarily irrational. But no human can live on strictly emotional decisions. Interestingly though marketeers understand that shoppers buy "impulse" items and market products accordingly.

Giving money to bums is not a market phenomenon. Besides you may be purchasing a Good Feeling About Yourself.

Your other questions are just as easily handled by adding more elements into your decision making function. Economics always starts with the simplest formulations of events by assumption then gradually relaxes those restrictions.

Rationality comes into the picture once a goal is established then you evaluate the means you are using to obtain that goal. Economic theory is built upon an understanding that individuals do not all have the same goals but all use rational means of obtaining them.