Can Quicksilver push bullets out of the way?

As far as the Quicksilver that appears in Age of Ultron, we never see him do anything like changing the trajectory of objects in mid-flight. The closest we come, I believe, is when he is able to snatch Hawkeye's arrow out of the air in the initial fight in Sokovia.

It's possible that he's simply not fast enough to do that. More likely, though, given how new his powers are to him, he may just not have the skill or practice to do anything that precise.

Note that there are two places in the movie where Quicksilver is struck by a bullet. In both cases, he is not trying to outrun gunfire. Rather, he runs into the trajectory of the bullet. The first is obviously an accident, when a friendly police officer fires into the area where Pietro is heading, and the bullet grazes his arm. The second time, he runs head-long into a hail of gunfire from the Quinjet.

I don't think we're supposed to believe he ran in front of every bullet to protect Hawkeye. Rather, he ran in and grabbed both Hawkeye and the boy, and carried them to safety behind a car. In doing so, he just happened to put his body in the path of the bullets. I suspect he knew there was a serious risk that he would get hit, but decided that saving the boy, and one of the Avengers, was a worthwhile sacrifice.

The muzzle speed for a aircraft is much faster that that of say the handgun Hawkeye used to make him fall through the floor. Think of it as catching a slow football toss vs stopping 10 footballs thrown by Manning from hitting a target

The closest we see Pietro coming to move objects at super speed is when he is punching someone or moving them out of the way of the train. But all of these people are moving at normal human speeds, not the high speeds of a bullet.

He does come close, when he disarms (ha ha, get it?) Klaue in the salvage yard, on the Churchill. Klaue fires a single shot from his Beretta 90-Two Type F - 9x19mm (As identified at the IMFDb), but Pietro runs in, and takes the gun completely apart.

Only 9 cartridges. This gun is using the standard single-stack 10 cartridge magazine. The higher capacity cartridges are double stacks and look different. 9 cartridges on the table, plus one fired. As Pietro is clearly trying to show off, IF he could have grabbed the bullet that was fired, or the spent cartridge, in mid air, to put with the rest, he would have.

There's also the less convincing scene of Pietro trying to grab MewMew mid air and failing.

He attempts it, so he clearly thinks he can affect it. A regular weapon he could maybe move. Mjolnir is not being thrown at super-sonic speeds, unlike the typical handgun bullet, let alone a fighter jet cannon. It's sufficiently slow compared to a bullet.

But He is also shown having some repercussions from hitting things or running. Out of breathe, hand/shoulder hurts. Unlike his counterpart, he isn't immune to the laws of physics. Equal and opposite reactions. Inertia.

But as can be seen in the video above, the Marvel/Avengers Quicksilver is SLOW compared the Fox/X-Men Quicksilver.

He can be clearly seen while movie. Certainly high speed compared to normal humans, but in the scene where Hawkeye shoots the ground out from underneath him, the handgun bullet is much faster. In the scene where he gets grazed by a cop at the end battle, he's clearly not fast. X-Men Quicksilver is so fast that he takes out multiple guards and moves multiple bullets before Wolverine could even finish popping out his claws.

Water/rain and objects thrown into the air are almost frozen in mid-air.

And that's him goofing off really. Well, not even. In the sequel, a ten year older Pietro is EVEN FASTER, while STILL GOOFING OFF.

Plastic Bullets and Sprinkler systems are snails compared to a high energy explosion.

In short, MCU Pietro can't move fast enough to affect bullets at muzzle or near-muzzle velocity. And probably would be affected by Inertia if he tried.