The German word »gern« usually means "to like". Your sentences are good examples for that.

Susi isst gerne Brot.
Susi likes to eat bread.

This means, that Susi often eats bread because it tastes good to her. The usage of "to enjoy" seems too strong for this sentence, because means to relish something, to have doing so, but for Susi eating bread is just normal. It is nothing extraordinary. It is something that feels good and positive, but she is not excited about eating bread. Eating bread is nothing she would mention in her diary.

Jürgen hat Irene gerne.
Jürgen likes Irene.

Irene is a person for which Jürgen has positive feelings, but he is not in love with her.

Ich hätte gerne eine Tasse Kaffee.
I'd like a cup of coffee.

This is just an order in a restaurant. This is what you want in this moment, but you have no idea if the coffee will be so good, that you would enjoy it.

Ich würde den Müll gerne jetzt runterbringen, denn später habe ich sicher noch weniger Lust dazu.
I'd like to take the garbage down now, because I'm sure I'll feel even less like it later.

Taking down the garbage is eben something that you don't want to do. But doing it now feels a little bit better than knowing that you still have to do it later.

But also:

Kurt vergisst gerne die Wohnung abzusperren, wenn er zur Arbeit geht.
Kurt tends to forget to lock up the apartment when he goes to work.

Kurt doesn't enjoy forgetting to lock the door, and he doesn't have fun leaving the door unlocked. He just often forgets to do so.

In feuchten Ecken entsteht gerne Schimmel.
Mold tends to grow in damp corners.

Mold has no emotions at all. It just grows where it finds acceptable conditions.

The German word »gern« usually means "to like". Your sentences are good examples for that.

Susi isst gerne Brot.
Susi likes to eat bread.

This means, that Susi often eats bread because it tastes good to her. The usage of "to enjoy" seems too strong for this sentence, because means to relish something, to have doing so, but for Susi eating bread is just normal. It is nothing extraordinary. It is something that feels good and positive, but she is not excited about eating bread. Eating bread is nothing she would mention in her diary.

Jürgen hat Irene gerne.
Jürgen likes Irene.

Irene is a person for which Jürgen has positive feelings, but he is not in love with her.

Ich hätte gerne eine Tasse Kaffee.
I'd like a cup of coffee.

This is just an order in a restaurant. This is what you want in this moment, but you have no idea if the coffee will be so good, that you would enjoy it.

Ich würde den Müll gerne jetzt runterbringen, denn später habe ich sicher noch weniger Lust dazu.
I'd like to take the garbage down now, because I'm sure I'll feel even less like it later.

Taking down the garbage is eben something that you don't want to do. But doing it now feels a little bit better than knowing that you still have to do it later.

But also:

Kurt vergisst gerne die Wohnung abzusperren, wenn er zur Arbeit geht.
Kurt tends to forget to lock up the apartment when he goes to work.

Kurt doesn't enjoy forgetting to lock the door, and he doesn't have fun leaving the door unlocked. He just often forgets to do so.

In feuchten Ecken entsteht gerne Schimmel.
Mold tends to grow in damp corners.

Mold has no emotions at all. It just grows where it finds acceptable conditions.

• Is it fair to say that gern + subjunctive expresses intention rather than enjoyment?
  Neither nor. It expresses a wish or request. The doctor wants to measure for fever. It doesn't enjoy him doing so, and it it not an intention, because it depends on the patients agreement.

• Is this construction used outside a medical setting?
  Yes

  Ich würde mich auch gerne noch wo anders nach schwarzen Schuhen umsehen.
  I would also love to look elsewhere for black shoes.

• Is this construction used with verbs other than mögen?
  Yes

  Ich könnte ihnen auch gerne noch ein Strafmandat ausstellen.
  I could also issue you another penalty notice.