First things first:
- Yes, reír can be used without the "reflexive";
- No, reír cannot be used transitively.
Now on to the details:
The use of the (formally reflexive) pronouns with verbs like reír or comer has been analyzed as a kind of mediopassive voice construction that conveys a sense of action of the subject for the subject's own sake.
Spanish grammar is presented traditionally as having two grammatical voices: active and passive, the latter being expressed periphrastically by the verb ser plus the passive participle of the main verb. There are also phrases like La puerta se cayó or Se rompieron los platos, which are not exactly reflexive, and can be termed a form of middle voice.
Finally there are phrases where the "reflexive" pronoun is optional, as with reír (yes, you can say Reímos mucho instead of Nos reímos mucho, though the latter is much more common). These are usually of a kind that allow a mediopassive meaning of benefaction for the subject. That's why you usually find reírse instead of just reír, and also why you'll find people saying things like
¡Me comí una pizza grande yo solo!
Tus amigos se tomaron toda la cerveza.
No te imaginas lo que nos compramos.
Fue de viaje y se trajo un montón de souvenirs.
In each of these examples the "reflexive" pronouns could all be removed and the sentences would still be perfectly grammatical, but they would sound either oddly distant or formal or missing the emotional touch that their context demands.
Things are not that clear-cut, of course (middle and mediopassive voice are somewhat blurry categories), but there you have it.
ADDENDA: I realized I didn't reply to the last part of the question, because it seems to be tied to reír as transitive verb, which is impossible, but I figure it could be useful to know how one might translate a "self-benefactive" mediopassive phrase of this kind into English. For reír(se) this doesn't seem possible. In Spanish the closest thing would be something like
Reí para mí.
Reí para mis adentros.
which translate as "I laughed [quietly] to myself". This doesn't convey the idea of reírse and indeed it may imply exactly the opposite.
For transitive verbs that denote acquisition of goods the emotional expressiveness can be expressed in English, as in Spanish, with a pseudo-reflexive, although in a much more general way:
Fui a la tienda y me compré el vestido más caro.
"I went to the store and got myself the most expensive dress."
Note that there might be a problem translating cases like the following, where the subject does something that benefits themself (emotionally) but that is intended for other people:
Se trajo regalos para toda la familia.
*"He got himself gifts for the whole family."