I have a caution from UK but not a UK citizen. I don't live in UK anymore. Will it effect my USA business visa application?

Do I answer NO for question "Have you ever been arrested or convicted for any offense or crime, even though subject of a pardon, amnesty, or other similar action? "

Well, the very simple answer to that is "answer YES" because you were arrested, even though a Police caution is an "out of court disposal" and not a conviction - you were still arrested...

To suggest that you answer any other way would be suggesting that you do something that would put you in violation of US law when applying for a visa.

If you want to see what the UK holds on your police record, request an ACRO Police Certificate.

Do I answer NO for question "Have you ever been arrested or convicted for any offense or crime, even though subject of a pardon, amnesty, or other similar action?

You have to answer YES, which is the truth.

Can they check my biometric of U.K. ? My UK visa is in my old passport and if I don't show the old passport they wouldn't know I lived in UK right?

Yes, they can check however it does not mean they will. Know that the USA and UK share immigration and law enforcement information more closely than any other two countries. See Five Eyes. Don't try to be smart or cunning. They have or can have access to more information about you than you know.

That said a simple crime of shoplifting does not necessarily make you inadmissible to the USA. From what you say, it is very possible (without more information) your offense may fall under exception (ii). You will however need all the disposition documents related to your arrest/charge to apply for the visa.

Read What are the chances of a US visa if I have a criminal record?

Ineligibilities and Waivers: Laws

(i) In general.-Except as provided in clause (ii), any alien convicted of, or who admits having committed, or who admits committing acts which constitute the essential elements of-

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(I) a crime involving moral turpitude (other than a purely political offense) or an attempt or conspiracy to commit such a crime, or

(II) a violation of (or a conspiracy or attempt to violate) any law or regulation of a State, the United States, or a foreign country relating to a controlled substance (as defined in section 102 of the Controlled Substances Act (21 U.S.C. 802)), is inadmissible.

(ii) Exception.-Clause (i)(I) shall not apply to an alien who committed only one crime if-

(I) the crime was committed when the alien was under 18 years of age, and the crime was committed (and the alien released from any confinement to a prison or correctional institution imposed for the crime) more than 5 years before the date of application for a visa or other documentation and the date of application for admission to the United States, or

(II) the maximum penalty possible for the crime of which the alien was convicted (or which the alien admits having committed or of which the acts that the alien admits having committed constituted the essential elements) did not exceed imprisonment for one year and, if the alien was convicted of such crime, the alien was not sentenced to a term of imprisonment in excess of 6 months (regardless of the extent to which the sentence was ultimately executed).