-The Senate gathers three systemic functions.
1) It is a tribune of the oligarchy. The richest and most prestigious men from Rome sits in the Senate and talks about current affairs. We can see it because even if the senatus consultum was broken by a plebs tribune, the talk was recorded. The important was to know what the other members of the oligarchy thought about the issue. This aspect is very important. If you are isolated thinkingsomething, you will not act on your own unless you are Marius or Caesar.
2) It is also a "Corps". A Corps is a group of people where all the member exert of pressure on each member to ensure its existence. Here, it is Warish affairs. The Senate had a routine (of war), they did war, they were prestigious, they just maintained themselves in this state because ... because it was the reason why their group existed !
This kind of "Corps" can be compared to the American Militaro-Industrial complex (military and cash-making routine),French "Corps d'Etat" (law-maker and statist routine), and English web of students from prestigious universities (Scom and prestigious pseudo-military routine). They do the politics on the long term beyond elections.
3) Finally, the Senate is the head of the city in times of crisis. Look at the Senatus Consultus Ultimum, which was a decree by the Senate. In order to "save the patria", anybody could kill the threat. Traditionnaly, a group of senators lead roman knights (equestres), scribs and tresor tribune armed with batons to kill the dangerous man.
-In practice, they were able to edit the laws passed by the people, but this edit could be broken by the intercessio of a plebs tribune, which could be broken itself by the veto of another tribune. These edits had no legal force in pactice but they were followed by all magistrates and citizens because of the auctoritas of the Senate. A Senatus consultum voted by a too incomplete Senate was usually ignored and crushed the day after.
-And how did you enter the Senate ? You had to be rich AND prestigious, you were excluded from the Senate by the censor if you were not prestigious not rich. But roman knights (rich people and prestigious too) were not automatically senators. You had to occupy a curule magistrature, that is to say one from the cursus honorum, questor, praetor, edil and consul.
-Who was influent in the Senate ? Before the Punic Wars, the one who were dictators once and ancient censors. They were the only one who truly talked, the other shut up and walked to the man they backed. After the Punic Wars, with the disparition of dictatorship, consulars (the one who were once consuls) became the dominant group of talkers.
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It could in fact meet without the magistrate. This condition was just a formal one, and the magistrate ran to the Senate when there were senators in it. You can read Marianne Coudry thesis "Le sénat, pratiques délibératives". Mommsen was wrong.
It could de facto declare wars. The Senate was the one who decided of international affairs.
Beware the notion of separation of powers ! It did not exist in ancient Rome. People in Rome had no specific filed of action. The Senate had particularly universal competence. They could talk and take decisionsabout narly everything ! The key notion for the Roman Republic is not the separation of powers but the actual balance of powers between three poles, The People (democracy), The magistrates (linked to monarchy for the Roman because they hold the powers the king formerly gathered), the Senate (oligarchy). The were all dependent upon the others. The people elected magistrats and gave its auctoritas to the Senate, the magistrates are the actual "men of action" in Rome (so that they are needed by both the people and the Senate) and the Senate is the assembly of the elite of the People and they can take many decisiosnabout magistrats and the People. But they need to be backed by the people and magistrats (who are senators-to-be) because they have no "legal" power.