If indeed the charges were dropped, no.
You can find the charge(s) they were arrested for on the arrest book of the law enforcement agency that arrested them, or the court charges on file at the Clerk of The Court office at the courthouse they were tried in.
If the charges are violent felony charges or felony drug charges you will more than likely be extradited to Missouri. If they are not felony charges there is almost no chance that Montana will do anything. Missouri must first file charges against you and then file a warrant for your arrest. Montana cannot file charges for something that happened in Missouri. They can only turn you over to the Missouri State Police to face charges there. I don't really know what "pick up charges on you" means, but you have to see a judge where the crime happened. You can be charged in Montana for something that happened in Missouri.
It depends on the job. Most employers don't run background checks to see if you've been arrested. They have to pay for that service. If you are applying for a job with a City, State or government agency you should probably say yes. Because they would check.
Generally speaking an investigation would not show on a background check unless it resulted in arrest. A background check for sensitive employment or government/police employment will probably show the investigation whether or not you were arrested.
You can be arrested, but it would be a difficult case to prosecute if the vicitim does not want to press charges.
Yes he was arrested for assault charges in 2002.
i just wanna see if i have any charges on my personal background
Yes and No. But not quite enough information is given to answer either way. Either way, if a police report was made, the report will always exist in the agency's files. If you were arrested, the record of your arrest will be on your record, BUT - so will the fact that no charges were filed. If you were never formally arrested, no, there will be no existing record.
These charges are USUALLY statutory misdemeanors and probably will not show up on a criminal background check.
File charges and have the assailant arrested.
getting arrested
It depends on what type of background check. If it is a criminal background check, yes, any charges should show up.