1.Can you use silver pieces for silver sling ammunition?
[When first writing this I'd not considered the mass of the coin compared to sling bullets. Both Dale M in his answer and Trish in her comment make good points that a GM might also... weigh when exercising their discretion.]
A. Yes, at your GM's discretion.
Improvised weapons are completely at the GM's discretion, per "Improvised Weapons" at PHB pp.147-148.
One can easily imagine a GM saying "it takes you a few minutes to mangle the silvers enough to fly as well as a stone, but that's not a problem." Or a GM saying "purchased sling bullets (PHB p.150) are akin to 18th-C. bullets: crafted materials that will do lethal harm at range. You can't just use any-old rock or coin." And there's nothing in the rules to tip the scales in one direction or the other.
B. Yes, with some downtime, and perhaps a little less discretion.
If you craft these silver bullets (PHB p.187), you can make a lot of silver sling bullets. Like, 2500 in a single day.1 Again, crafting requires a lot of work with your GM, but I think there's a much better case to be made that even one untrained in weaponsmithing could mangle coins into pretty-good ammo-shape given a day and a hammer.
You could even call the first hundred your "practice bullets," then re-craft those raw materials, selecting the best twenty to put in your pouch, in under an hour!2 All for just the cost of 10 sp.
[Note: there's nothing actually in the book about crafting over "partial" days. Union rules may require you to work the full day.]
2.What attack bonus/damage options would exist?
Whether or not this ammunition is similar "enough" to "real" sling bullets to allow you to exercise proficiency is, again, completely at your GM's discretion per "Improvised Weapons."
On the damage front, you've lucked out: improvised weapons do d4 damage unless it "is similar to an actual weapon and can be treated as such." But your sling bullets already do d4 damage, so either way the GM rules you're looking at d4 damage.3
Lastly, there's nothing in "Improvised Weapons" nor in Sharpshooter to rule out sharpshooting with these coin-bullets. Unless, as already discussed, the GM rules that you're not proficient in mangled-coins-in-twirled-fabric-straps.
1 - That's 5gp-worth of crafting divided by 4cp/20 bullets, for 2500 total bullets, given the input materials. Of course (assuming one coin per bullet), if you actually had that kind of silver at hand, you'd not need to ask this question. So you'll get to take a long lunch and clock out early, probably.
Again, I had not originally considered mass; If you like Dale M's answer these numbers would need to be adjusted by a factor of ~4.
2 - So that's 200 bullets crafted (albeit destroying 100 and rejecting another 80) for a total value-created of 40cp, or less than one-tenth a crafting day.
3 - Unless the GM decides, as is completely their prerogative, that these improvised bullets should have a different damage. Like, say, 1 or 1d2.