What serious problems did the Republic of Texas face as a new nation?

The Republic of Texas was sparsely populated, the Native American population was hostile to white settlers, and the white settlers themselves were hostile too, leading to social conflict.

Besides, the Republic enjoyed very little international recognition, had Mexico declare war on it, and most of the population wanted to merge with the United States.

These factors ultimately led to disappearance of the Republic and its merging with the United States.

in all honesty, it really hasn't. except that hitler killed himself.