Why my old emails show up when I use the spotlight search?

because they're still part of a cache that Spotlight can search. Eventually, as the cache is overwritten, those emails will cease to show up. If you don't want them to show up right now, you'll have to restore your phone as a NEW device not using ANY backup at all.

Hi ..

Try a reset.

Press and hold the Sleep/Wake button and the Home button together for at least ten seconds, until the Apple logo appears.

If that doesn't help, tap Settings > General > Reset > Reset All Settings

No data is lost due to a reset.