Have JWs always been politically neutral, or did they in the past e.g. vote in elections or support one party over another?
The majority of history I can find about Jehovah's Witnesses having political neutrality is regarding being conscripted for war. Specific dates are hard to come by, but war neutrality was first widely encouraged in 1904, followed by another refinement in the Sep. 1, 1915 Watchtower, and then finally adopted completely with the landmark article titled "Neutrality" in the Nov. 1, 1939 Watchtower. (Source)
The Bible Students, now known as Jehovah's Witnesses, knew from early on of John 18:36, which discusses how God's Kingdom is not part of this world. I can only speculate, but the consciences of those early Bible Students probably kept them from needlessly involving themselves in politics. Most controversial standpoints at that time were about how to react to compulsive military service.
Does this political neutrality thing mean that JWs do not meet with U.S. Presidents or other political leaders to discuss policies, or have such meetings ever happened?
Jehovah's Witnesses don't discuss politics with anyone. They're completely neutral for the sake of giving exclusive devotion to God's Kingdom. This is advantageous both for the purposes of ministering to "people of all the nations" (Matthew 28:19) and for the purposes of having unity within the congregation (Psalm 133:1).
JWs do assign much respect to local authorities, and would not refuse meeting with the authorities, but they would not give opinions on politics or promote a political campaign.
Does it mean that JWs do not sing national anthems such as "Star Spangled Banner" as part of their church services?
Correct. At football games, baseball games, etc. I usually excuse myself during the beginning of the game and go walk around in order to avoid making a scene, but during these ceremonies some JWs will remain seated, others will perhaps stand up but not put their hand on their heart. This isn't meant to be seen as defiance, only as non-participation.
Likewise, in all 13 years I went to public school in the United States, I would just stand respectfully during the Pledge of Allegiance. People very rarely asked me why.