I just test drove a 2007 Toyota Corolla CE with 86k mileage and no codes/dtc with a 5-speed manual. (After a couple of times of stalling the vehicle), I noticed that the acceleration in first and second(less with second) gear were not smooth, with the vehicle constantly jerking back and forth. I was wondering if this is a common problem with manual vehicles or not.
No, it is not common or normal for a manual transmission car to do that. I have one over 20 years older than that and it accelerates smoothly in all gears. If the engine itself is running OK I'd suspect bad engine mounts or maybe a CV joint problem.
The engine takes a slightly bit longer to start up with a little bit more movement back and forth. a bad motor mount?
Could be. Worn engine mounts are a common issue on older vehicles.
https://www.youtube.com/watch?v=g9GNzeK0wRU