First, please do send your child to preschool. The incidents you describe are in mommy-and-me types of classes and situations where parents are the authorities present (despite the presence of "teachers" or "leaders"). Things in a structured classroom setting with an experienced preschool teacher are very different, unless it's a very bad preschool indeed. Keeping your child out of a good preschool program out of fear she'll be hurt is unrealistic and does her a disservice -- a good preschool will prepare her well for kindergarten and school eventually. And in a preschool that's worth its salt, a child who bites or hits or kicks or shoves is a child who is watched closely and corrected - and who is ejected if he or she can't get with the program and stop it. You are basing your fear on a few incidents that sound frankly very typical for these ages.
I also am a bit taken aback at how very, very detailed your descriptions of these incidents are. While it's great that you're paying attention to these other kids' misbehaviors, I also would recommend that you read some good books on "ages and stages" to learn what is typical and normal behavior for children in the toddler stage, and up to and including age three. "Normal" does not always mean "nice" -- which you'll realize if you read up on development and behavior. Yes, in the two incidents you describe, the kids should not have behaved as they did, but an adult should have intervened before the kid shoved the chairs together, and in the first case, one shove resulting in clunked heads -- well, it's hard to stop that unless you're standing right over the kid in question. The child was corrected and the adult apologetic. End of issue unless the kid then roamed the room doing it again and wasn't removed.
Did the shoving kid then go off and shove everyone around? Repeatedly? Did the kid who pushed the chairs then keep on pushing and run off and do something else "violent" once he was stopped? That would worry me, but single incidents would not. I would worry if the same kid, every week in class, was rough. But again, one incident, corrected? Typical, not a reason to assume violence.
I also note that you seem to think that the child who "was about the size of a three year old" should have known better than to shove the two-year-old. Some kids are huge at two; that kid may not have been three at all. You also note that the child who shoved the chairs is three (for sure, or are you guessing based on size?). If you think that kids of three are supposed to know not to push or shove kids who are two -- please read up on development! The post seems to indicate that these kids were intentionally choosing to attack other kids--as if they planned it. At this age, they simply do not have the mental ability to plot things out intentionally. At this age it's about "I want X, you are in my way NOW." And it happens in a nanosecond.
I am not defending these kids' behaviors. But I also don't see this as some trend toward "violence" as if three-year-olds (assuming they even were three) should be far better at self-control than two-year-olds. Yep, they are older and more developed. But that doesn't mean they have the ability to control themselves ideally.
Consider the situations and not just what you are seeing as the kids' intentional violence. These were group gatherings -- "multi-age classes" as you note. Kids feed on other kids' energy and group play can get out of hand, which requires removing the child for a few minutes or even entirely. If these classes have an "anything goes" vibe and the parents are off to the side chatting, expecting a class leader to do it all, then find other classes instead where things are mellower and better supervised. Not all classes are like this. And not all older toddlers are out to get younger ones. It's the age and stage (and a lack of parents being close enough and watching closely enough much of the time, possibly). But don't deny your child preschool based on this.