Is it possible to make an output only spendable to a specific address? Basically, the entire question fits into the title. Does the Bitcoin scripting language allow for a transaction output to only be spendable to a specific address? Is it possible to forbid all other spends? Is it possible to dictate that only that spend over a specified amount of money can happen such that the transaction fees cannot be chosen arbitrarily large? 

If so, I suppose it's also possible to design an output script in such a way that it can be spent to a specific address (and only to this specific address) when a signature by a specific party is present, but to arbitrary addresses if that party's signature and also that of a specific other party is present. Should this for any reason not be possible, please let me know. This is what I'm mainly interested in.
As far as I know, the answer is no. 

What you are suggesting would imply that an UTXO could affect the conditions in which the output of a transaction spending it would be built, but it can only affect how the input is built.

UTXOs can define how they should be spent, but no how the resulting UTXO spending it will be created.