Not necessarily. Solid lead bullets were used up until about the 1890's IIRC. I'd be better able to tell with a picture that didn't have as much light on it while still being close up.
If there are little marks all around that groove like it kind of looks like, those are usually tell-tale signs of crimping, which didn't occur widespread until the 1880's or 1890's, when brass cases became standard instead of mainly for military use. Also, there aren't very many varieties of single groove bullets with a rounded dome tip that were used during the Civil War.
Still not sure on the brass piece. It seems familiar to me but don't know from where. Still has me

. Also, I don't think the other piece is any part to a trigger, or trigger guard. It looks like it may be a part of a frame buckle though.
If you're interested in the relic side of the Civil War, I'd recommend getting the following books for starters:
"Civil War Artifacts" by Howard Crouch
"Civil War Projectiles 2" by W. Mckee & M.E. Mason
These two cover a large portion of any Civil War artifacts you'd be finding starting out. Past that it's just time, patience, and experience!