Wow. I wanted to say smth like "g++ already has int128", but CF has only 32-bit g++
Thank you, author =)

If there are any bugs or optimizations

There are. I will comment only IO-part. Marks TL / WA mean type of comment.
Spoiler: nothing very important.

TL void putStr(std::string s)

Use const &s, do not make copy of s.

TL putStr(solve(p,q,b).c_str());

You convert string → char* → string :D
Any way, it is useful to have also void putStr(const char* s)

TL void putInt(T number) { putStr(std::to_string(number)); }

If you wanna be really fast, do not convert to string. At least, you create one extra string object.

TL static char buffer[1024*1024];

If you make buffer less (16K), caching will speed up in some cases.

WA T getInt(): return number * sign;

It leads to undefined behavior if you try to read int =  -2³¹. About signed integer overflow.

WA void putChar(char c): if (size == 1024 * 1024 || c == -1)

Nice solution to flush buffer. Note, char c = -1 is rare but correct symbol ('я' in win1251).
So it's better to make void putChar(int c)