Differential equation questions I am studying differential equations of order $1$ and $2$ and I had these questions on my mind:
1. Is it true that every differential equation has infinitely many solutions if there are no initial conditions specified?
2. Is it true or false that the solution of every $2$nd order differential equation always has $2$ independent constants?
Thank you for helping me out :)

No, but these statements are false. Consider the first order equation
$$f'(x)^2+f(x)^2=0.$$
It is easy to see that even without initial conditions, it must be that $f(x)=0$ identically. For your second order question, similarly consider
$$f''(x)^2+f'(x)^2+f(x)^2=0.$$
You should generally be wary about statements that claim to hold in all cases without many assumptions.
If you would like to include complex valued solutions then replace $f$ by $|f|$, $f'$ by $|f'|$, and so on.
