What you are looking for is The Central American Seaway. There has been alot of research done on how this body of water and the creation of the Isthmus of Panama, roughly 5 million years ago, affected the world's climate. It was a long time ago, so scientists still haven't figured it out exactly.
This is an excellent summary and critical review of how the closing of the central american seaway may or may not have resulted in the Ice Age. Closing of the Central American Seaway and the Ice Age: A critical review you will mainly be interested in the Abstract, Introduction and sections 5 and 6.
Again the spectrum of responses denies us, at least at present, a definitive statement of how an open seaway would affect ocean circulation.
This is a rough idea of how the ocean currents would have looked
With the Atlantic and Pacific Oceans connected directly there would have been an even exchange of water. This would have affected the salinity of the water and the strength of the currents. I recommend this article explaining how the closure of the Panama Isthmus could have affected climate, especially in the Northern Hemisphere.
•How could this affect the way the ocean currents would move?
It would weaken the strength of the currents and therefore the direction they take. It would affect the salinity which would impact on the oceanic thermohaline circulation system (which, amongst other things, relies on the denser saltier cold water in the poles to sink).
•Would there still be the same oceanic gyres?
•Would we still have the gulf stream?
Not likely. A very weak gulf stream may still exist but it definitely wouldn't be able to bring such warm weather to the northern hemisphere that it currently does. It would be a lot colder up in Europe. The gulf stream would also possibly flow more southwards towards Spain and the Mediterranean area rather than up toward the UK and Scandinavia.
ie. if the thermohaline circulation system collapses or exists in a weak form (as suggested in section 5 of the first link I gave) you pretty much have....
BUT these simulations were based off current weather and ocean patterns in relation to the central american seaway RE-OPENING! Like the paper stated at the very beginning, we don't know what the ocean circulation was BEFORE the closure of the central American seaway.
•Would this impact on the El Niño climate cycle?
Definitely. Roughly 3-5 million years ago, there was a permanent El-Nino like state compared with the modern day El-Nino Southern Oscillation (ENSO). This change from permanent to oscillation could be attributed to the closure of the Central American Seaway.