Are the Police Allowed to Stop and Arrest Me Without a Warrant and Conduct a Search?

Generally speaking, no, unless you give them consent, and I do not advise you give them consent. There are exceptions to that, however, such as whenever they witness a crime or if they have probable cause to believe that a felony has been committed. In that case, they could actually arrest you without a warrant, but the general rule still stands that they need a warrant, and unless you give them consent under other exigent circumstances, they cannot stop and search you.