Road bike (commuter) frame size problem

I am roughly your size (same inseam 5'8") and 52cm for a "standard" diamond frame bike would be too small for me. With a long enough stem I could get the bars in roughly the right place, but it they would likely be too low without some serious bogdery with stem riser extensions. But if you're young and flexible enough to deal with the low bar position it might work for you.

However, 52cm doesn't really tell us enough about the exact frame geometry to know whether this bike is appropriate or not. It's close enough that you can likely ride it safely if not as comfortably as possible. A picture or link to the exact bike would help a lot in answering this question.

A bike only becomes "unsafe" when you can't get the proper seat height w/o having enough seat post in the frame seat tube. There are markings on the seat post ( generally around 4 inches). Or you can't pedal the bike w/o your knees hitting the handlebars ( generally this is when standing to pedal).

Beyond the basics bike sizing is mostly about getting the compromise of comfort and efficiency in fit that works for you. If you were a pro bike racer, 52cm might be ideal. For the average rider, you can likely make the bike "work", but you'd be more comfortable on a slightly bigger bike ( either 54 or 56cm depending on exact geometry.)

Q1: If it not then is there a possibility to get injured?

A1:

Too low saddle height will usually hurt knees and it is the first element to adjust.

It should be no pain or soar or any uncomfortable to go riding bicycle. And, if you feel any uncomfortable, it is then a signal to check and examine.

If you feel that, you can ask your physican or post the issues with details and pictures here.

Q2: Is it the right size?

A2:

I am the same height with you 175cm/5"9' and I have many road bikes. One of them is 52" and I have also 54". And, I have no problem with my 52" at all. The longest mile I rode with 52" was 360 KM or 223 Miles. I have ridden 52" for more than 15,000 KM in last 3 years.

I tend to ride a smaller one. A slightly smaller bike is more capable to adjust and fit rider's body more easily.

The key elements to adjust a smaller to fit rider's body are

1. Saddle height
2. Seat poistion - Fore or aft
3. Stem length
4. Stem height
5. Handlebar angle
6. Crank length

Here are few good videos to check.

Go checking them and others, do the fitting and good luck.