How did early Rome mobilize?

While this whole era – before the Third Samnite War (298 to 290 BC) – remains a bit in the dark for Rome itself, and data for this regarding its allies are even harder to come by, some of these are described en detail by a nice primary source: Polybius in Book IV (VI. The Roman Military System). Although he wrote at a later time, opinion goes that if his descriptions are reliable he has to describe a mixture of contemporary to his own recruitment patterns and older customs. The caveat: Some of the details, regarding the numbers, presented by Polybius are contradicting other sources, some are plainly impossible. Especially compared to the later Livy, whose dependence on Polybius makes for a very interesting comparison between the two when they differ.

Livy’s account must be largely derived from much later sources, especially Polybius, so that its independent value is not great. Yet its very incongruities may lend it a certain measure of authority. Livy may have been attempting to reconcile patchy and discordant source-material; but it is difficult to suppose that the legion he describes ever existed as a reality.
_{From: Lawrence Keppie: "The Making of the Roman Army. From Republic to Empire", Routledge: London, 1984.}

These difficulties in interpreting the sources have to be considered when reading Polybius:

When the consuls were about to enroll soldiers, they announced at a meeting of the popular assembly the day on which all Roman citizens of military age had to present themselves. On the appointed day the popular assembly or the consuls divided the 24 military tribunes into four groups, corresponding to the four legions that were going to be created. The distribution of the foot soldiers then took place on the Capitol on a tribe-by-tribe basis, with the recruits being led forward in groups of four and the officers of the four legions being given the right of first choice in turn.
_{From: Luuk de Ligt: "Roman Manpower and Recruitment During the Middle Republic", in: Paul Erdkamp (Ed): "A Companion to the Roman Army", Blackwell: Malden, Oxford, 2007, 114–131.}

The individual sub-questions:

1) Who was responsible for the mobilization? Was it simply the consul who was to lead the army, or were there special officials responsible for it?

A tiny bit nonsensical as this is phrased: the consuls were responsible, but of course, they could not do this all by themselves, hence they needed other "officials" to support them, starting with the military tribunes.

2) How exactly were the citizens divided into lots? Did they simply take the tables of the last census and raffle until the required numbers came together? Or were the requirements handed down to tribes and centuries?

That was indeed subdivided early on and nicely compartmentalised down to the local communities:

From each tribe they first of all select four lads of more or less the same age and physique. When these are brought forward the officers of the first legion have first choice, those of the second choice, those of the third third, and those of the fourth last. Another batch of four is now brought forward, and this time the officers of the second legion have first choice and so on, those of the first choosing last. A third batch having been brought forward the tribunes of the third legion choose first, and those of the second last. By thus continuing to give each legion first choice in turn, each gets men of the same standard. When they have chosen the number determined on — that is when the strength of each legion is brought up to four thousand two hundred, or in times of exceptional danger to five thousand — the old system was to choose the cavalry after the four thousand two hundred infantry, but they now choose them first, the censor selecting them according to their wealth; and three hundred are assigned to each legion.
_Polybius

So, this is important: lottery was rotated through the districts or tribes, but there this game of chance ended then. Men were elected/chosen according to their status as fighters, experience, age, number of campaigns etc.

The division and appointment of the tribunes having thus been so made that each legion has the same number of officers, 2 those of each legion take their seats apart, and they draw lots for the tribes, and summon them singly in the order of the lottery.
_Polybius

Finally:

3) Were there any exemptions? For example, having served in the last campaign? Being newlywed? Being the last descendant of notable family? Being under arrest?

This is not easy to answer in all its details and variations. Draft dodging was certainly possible to a certain extent. But it also made you unpopular in your community or even lead to your exclusion from it. This would likely be more severe than the punishments from the official side. The newly-wed case is largely irrelevant since campaigns in those times usually started after the seed planting. Having been in the last campaign and having survived it would in all likelihood rather increase a recruit's desirability than decrease it. But even more important is the fact that you were forbidden from running any kind of political office before completing at least ten campaigns and a certified legal right to decline further campaigns was only granted after 20 campaigns.

The system was designed to distribute the burden of duty and chances of fame fairly equally. The desire to not go to war seems to have been relatively low, while the higher ranks of society, the notable families, were even being also very eager to distinguish themselves in battle.