Most certainly not. "I already agreed with him" means "Я уже с ним согласился".
"Договорился" implies some form of negotiation or interaction.
"Согласился" does not imply anything other than than a nod (or even a mental agreement with someone else's opinion). Neither does "agreed" without any additional qualifiers.
So, no, "I already agreed with him" is in general a poor translation of the original Russian sentence. It can work in some contexts, but it is most certainly not " the only thing that should be accepted".
Not quite. Russian "договорился" сan mean "to come to an agreement" (formal or not) or "to agree on something" (but not with something - that's where English "to agree" fails as possible a translation). E.g. you can use Russian "договориться" to describe reaching an agreement on a meeting place while planning to go out. Certainly no paper agreement is implied here.
Russian "Я уже с ним договорился" may or may not imply any business dealings or reaching a compromise, which is how I would interpret "We've made a deal". Its meaning is much broader. You can use the Russian sentence to indicate that the two of you have agreed on a pub where you plan to watch a game tonight. "Making a deal" can hardly be used in this context.