1st you CANT battle random people
2nd you need their names and Friend Codes
3rd ill battle you if you whant just send your FC and name
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Trade with other people who have heart gold or soul silver.
to get bp (battle points) you have to beat people in the battle tower.
It's not the "last" Legendary Pokemon that you can find in Pokemon HeartGold, but most people usually catch that one last given that you need to have both Kyogre and Groudon in your party for Professor Oak to give you the Jade Orb to get to Rayquaza.
Yes you can locally trade between Pokemon Pearl and HeartGold meaning the people that you're trading with must be in an area that is close to you.
Yes it is possible but it is not possible to battle people with ds from your computer and i prefer pokemon gold not silver