If I want to make a grilled cheese sandwich with cream cheese instead of cheddar, can I get it melted like cheddar?
No. Cream cheese will simply go runny if you melt it. It is in no way a substitute for a hard cheese.
Cream cheese sandwiches are tasty, when prepared properly, but you cannot grill them the way you grill grilled (hard) cheese. But you don't need it to melt - it's already nice and malleable, and has the oil in a soft enough state that it hits your tastebuds properly.
Some good options for cream cheese sandwiches (or any similar soft cheese, really):
Toast or grill the bread as if for grilled cheese, but leave off the cheese. Then when it's done, let it cool just briefly and apply the cream cheese. Top with a flavoring agent, such as dill, parsley, etc.; very similar to toppings for a bagel with cream cheese. Can be eaten open-faced or closed.
Eat with cold bread. This is what my wife often does: just put cream cheese between two slices and eat. Works well with thinly sliced relatively bland bread, for a "comfort food" effect. Not all that flavorful, but for some it's a reminder of being a kid.
Cream cheese and lox sandwich. Or any strongly flavored cured meat, really. Prosciutto, for example. Yum. Again, bread toasted or not toasted at your discretion.
That sounds... unpleasant. Cream cheese will melt, yes, but it won't melt in the same way as cheddar. It has more moisture and a very different structure. Where cheddar will retain some structure and a "stretchy" texture, cream cheese will just flow more freely. Layered between bread, it would probably just squirt out of the sides of the sandwich when you bit in. The extra moisture would probably also be partially absorbed by the bread and leave the sandwich soggy.
If you have tried to spread cream cheese on a hot bagel, you can see how readily it melts and oozes out the sides. It wouldn't make a very good grilled cheese. As the above answers indicate, spread it on bread, toasted or not. I love it with raspberry jam in there, too.