Hi, Hopefully this will help and not be unclear due to language barriers, English is not the language i usually work in.

The armies raised by the senate spans a rather large period, but for this case I assume that you mean the republican era, pre marian reform (107 BCE) and post the Manipular legion. The legion of this period is generally known as the Polybian era. And this is the army most often portrayed in Videogames (Total War etc.) and the one who fought the Punic wars against Carthage. Polybius is our main written source material and his series Histories are a goldmine for Roman arms and their organisation during the period. Livy is also very useful. As always the ancient authors should be handled carefully and it is always good to approach them with a critical mind.

The information that you are looking for is in Polybius Histories 6 chapter V & VI. Chapter 5 deals with the roman constitution and nr 6 with the roman army and arms. If you need details these are easy to find on your preferred search engine. But I will summarise to answer your questions in short.

Who were responsible for the tasks?

The Responsibility for assembly of the legions was the Consuls. The Romans appointed Two Consuls per year, and they carried almost absolute power for their terms. Only the Tribunes were exempt. They chose officers for the army and commanded it in the field.
See. Polybius Ch 5 Section 12

A Consul during normal circumstances led two legions (16-20.000 Infantry and 1.5-2500 Cavalry). Circa half of the legion was composed of roman citizens and half consisted of allied troops. After being picked by the senate the Consuls also appointed 25 military tribunes and they in turn were assigned to the legions in pairs. The tribunes for each legion then picked the men they wanted for service and also appointed officers for them.
See. Polybius Ch 6 and Greece & Rome at War, Peter connolly 2006 p. 129.

Who forged and provided the equipment?

The Equipment used by the troops during the Republican era was bought by the soldier for his own money, and followed certain guidelines. For example the Hastati (First line infantry, Lightest armed and Youngest in the heavy infantry) should have: Sword, 2 Pila, Helmet, Scutum, Greaves and a plate on the chest called Pectorale. If the man has a rated income above 10.000 Drachma the Pectorale will be exchanged for a chainmail (Hamata).
See. Polybius CH 6 Section 22 & 23

On the production little is written and the finds of arms from the period are not many. I have not found anything tangible in my sources for the Republic era (Polybius, Goldsworthy or Connoly). The weapons and armor is probably made locally and to the specifications of the buyer.

There is a lot of material to read and good pictures if you need an idea of the equipment the soldiers used during the period. The Polybian legion in a standard search should land quite a few hits.

On the Question of time for preparation:

The arms were owned by the soldier and part of his household. So the army during this period did not give arms form stockpiles or workshops to the troops. The soldiers were as part of their duty to the state be ready to defend it. Thus troops could potentially be mustered from the entire pool of available manpower at a specific time (determined by the census performed to ascertain a man's wealth and place him in the classis system determining his place in the army).

During the Punic wars as Hannibal Barca entered the po valley rome had forces penned in at Placentia and Cremona, During the winter of 218-219 BCE (after the battle of the Trebbia). The Consuls for the year raised new troops in response to this defeat and had two new armies ready att Rimini for the consuls in Early spring. Thus two to four months is for this case the time it took for the roman state to deploy an army from scratch from Latium to the Southwestern Po valley.
See. G&R At War Connoly p.171-172, Livy, Periochae 21-25

**Abbreviations:**BTE: Before Common Era (Alternative to BC)

Litterature:

Connoly Peter, Greece & Rome at War (2006)

Goldsworthy Adrian The, Complete Roman Army (2004)

Primary Sources:

Livy, Periochae 21-25

Polybius, Histories Book VI