With the beginner's set, there are no examples of droids other than the 4th level battle droid (I have no clue what this means) that is 41-VEX..

Any idea how to deal with other types of droids? I'm thinking of the R2 series (astromech) and other useful droids. Let's say that a group wants to get or rebuild a parts droid? Any idea on how they could use it?

How to alter the programs to improve say "Astronavigation", or use the droid as a repair droids (Mechanics) for their ship?

There are 5 classes of droids in the Star Wars galaxy. 1st class droids are basically built for knowledge, science and medicine. 2nd class droids are built for technical and mechanical tasks, 3rd class droids are service and protocol droids, 4th class droids are built for combat and security, and 5th class droids are built for manual labor. So the pre-made character is a combat droids, although he has specialized in medicine (so he's like a combat medic). In fact, in the Edge of the Empire beta, 4th class droids are the only kind of droid you can pick. I'm just guessing, but I suspect there will be other kinds of droids in the other editions of this RPG.

The way the game is built, once the Core Rulebook comes out, you can use this type of droid for all kinds of different purposes. You can be a Bounty Hunter, Explorer, Hired Gun, Technician, or Smuggler, each one with three different Specializations.

The pre-made character is a Colonist basically, with a Talent Tree that combines all of the three specializations for that career (Doctor/Politico/Scholar).

Since only the 4th Degree droids are really allowed to do what most players want to do, they are the ones that FFG included in their game. For others, like R(1-5) or Protocol, which are really only non-combat types (yes, R series droids can effect repairs in combat, but can't do any fighting themselves), so they'll probably just be gear in the Core book that you can buy/upgrade.

The issue is in the tone of the game. This is not a survey of the entire Star Wars universe but rather a look at the scum and villainy that sits in the shadow of the empire. Considering the legal status of 4th Degree droids it makes sense they would tag along with the likes of smugglars and bounty hunters (The ever so famous IG-88 and HK-47 are perfect examples of this). There's also a technical problem: while there are skills included with the book to help flesh out a plethora of droid types (an astromech with Pilot to assist and Mechanics to operate, a 1st Degree with an astonishing Medicine skill), for a PC the classes are NOT really geared towards those kind of specialists but rather explorers that dabble in these trades. For player characters the 4th Degree is the best option to help the overall story, I feel, as something like a protocol droid might not be as usefull in all situations (unless playing C3PO is your thing, I guess). The 4th Degree droids represented here are sort of jack-of-all-trades when compared to other degrees of droids purely out of a neccesity of survival. At least that's my take.

Now for NPCs it's a completely different story. You can easily represent a droid of any other type by using the quick NPC rules of assuming 2 ability dice (more if you feel they're extra special) and ranks in particular skills you feel are appropriate. So like I said an Astromech could be decent as Mechanics and some other tricks, but it wouldn't make for the same kind of progression and character arc that the PC droids in this game are fleshed out to be.

Maybe in one of the other two core games to come out will include other degrees of droid instead of the 4th Degree found here.

The astromech in our campaign has been with us for so long he is becoming his own character. What our mechanic has done are a number of hacks and upgrades to increase his versatility. Not with out certain cost by the time out GM has had his say but definalty worth it. I think there is a definite noche there and we have been fleshing it out much the same way we have modified and enhaced our ship. just saying.