Is my prayer valid, and am I obligated to make up 21 days of prayers, because I used to perform ablution while having a splint on my hand, and I did not perform Tayammum?

It is permissible to wipe over a splint (or cast), provided it does not exceed what is necessary. If it does exceed what is necessary, and there is no harm in removing the excess, then it is obligatory to remove it and wash what is underneath. However, if removing the excess would cause harm, then it is permissible to wipe over the entire splint.