Does Republic of Rome usually generate one or more wars per turn, leading to a loss even when cooperating?
You really do need to read the rules, more thoroughly than it sounds as if the rest of the group has done. If you do, you will find that RoR is not a common-or-garden multiplayer game where an ally is somebody you have no reason to attack at present; it is designed so that the players all have to genuinely co-operate to keep the state going, and the winner is the one who can use each successive crisis to best advantage. The designer's notes say "I designed it for those who find Diplomacy too tame", and you need to be a born politician to find the game easy. But the notes also include a page or two of suggestions, and here are one or two of my own.


The object of the game is not to preserve the Republic; it is to have yourself declared Emperor (or some near-equivalent). One of my favourite winning conditions is being in rebellion when the state goes bankrupt; presumably the Ancient World Bank then puts you in charge of rebuilding, Monti-style.
The Early Republic scenario is well-known to be the hardest to survive; if you don't like it, try the Middle or Late Republic scenarios, or even the optional one where you shuffle all the cards together and use the top fifty or whatever. But bear in mind the reason (mentioned in the rulebook) that not every year is represented by a turn: a year when nothing happens is a year when nobody has a chance to affect the game. If you are not versed in the subtle politics going on around you, a game without wars is going to seem quite dull - until someone else suddenly wins. 
There are precisely six random events each turn, and a lot of possible wars. When you include the enemy leaders then yes, you are going to have to do a lot of fighting; at least one war a turn. You can't afford to say "I'm not voting to give that general a chance to shine", or "I'm not contributing to the treasury unless everybody does"; rebuild the army, and send it out to fight.
Setting the terms of the debate is a huge advantage: if the HRAO (think President) has done his job, you have an unpalatable choice between everybody losing and making someone else's victory easier. But of course you may be HRAO yourself next turn...
You need to think quite seriously whether you would rather have somebody else win or everybody lose (see this question). Of course, if you are winning, you need to consider that the other players are also thinking this, and won't give you an easy ride. A lot of Early Republic games end in one climactic battle; if the army loses, the state collapses, and if it wins, the general has enough prestige to win the game. But he might be killed while winning the victory, which would give others a chance, so it's worth voting to send him out; now, how many legions do we send to maximise the odds? Unless you think this way every turn, you won't be a good RoR player; be consoled by the fact that you are probably a nicer person for it.