No You dont
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When you first get to Kanto, you will see it in the water after you get off the S.S. Aqua. Then you will see it again outside of Fushia City (and after crossing or going to Silent Bridge) at which point Eusine will say he's seeing a pattern and will tell you he likes watery hilly areas in the north. Then, you fly to cerulean city and go north, and there you will find Suicune at level 40. Happy hunting!
The only way is to beat the elite four then check again. Same with other Pokemon, like if you beat suicune and then beat the elite four, suicune will be at the burnt tower, the place where you first saw it, and you can try to catch it again. It only works if you killed the Pokemon though, i know cause i tried to get a second lugia.
beat the league again and it will be roaming again
It depends on the Pokemon game you're playing. In the 4th generation and later games, you can re-encounter Suicune if you defeat the Elite Four again, while in earlier games, you won't be able to encounter it again.
yes but you have to beat the elite 4 and ash again