Is it permissible to perform tayammum for prayer when there is a splint on half of the hand during janabah, and is ghusl obligatory despite the presence of the splint?

If you intend to perform ablution (Wudu') or a ritual bath (Ghusl) and you have a splint (Jabeerah) on you, wash all the sound parts of your body that require purification, and wipe over the entire splint. This will suffice you. If the splint extends beyond the necessary area for its tightness, then you are either required to remove it and reposition it only over the necessary area if you will not be harmed, or to wash the sound parts, wipe over the splint, and perform Tayammum for the excess area if you fear harm from its removal. This latter opinion is the more cautious one. However, some scholars permit wiping over the splint even if it extends beyond the necessary area, to alleviate hardship.

If the splint was placed without prior purification, wiping over it will suffice you, and you are not required to perform Tayammum in addition to it. This is the preponderant opinion, though performing Tayammum as a precaution is preferable.

Wiping over splints is permissible according to the majority of scholars. It is reported from Ibn 'Umar (may Allah be pleased with them both) that he used to perform ablution over a wound on his finger. Wiping over splints is done with water, not with dust (Tayammum). Most scholars stipulate that for the wiping to be valid, the splint must have been placed while in a state of purity, and it must not exceed the necessary area. However, Sheikh al-Islam Ibn Taymiyyah holds that prior purity is not a condition, in order to alleviate hardship.