Is it permissible for someone who experiences a wet dream (ihtilam) and is unable to bathe or perform ablution due to quadriplegia to pray and fast, particularly since his elderly mother cannot assist him, and he only performs tayammum (dry ablution)?

It is obligatory for you to perform ghusl (ritual bath). If you are unable to perform ghusl yourself, then seek assistance from someone who can help you, even if it requires payment. If you are unable to do all of that, then you should perform tayammum (dry ablution), and your prayer performed with tayammum will be valid until you are able to perform ghusl.