I can't comment on any specific type of armor as that is too detailed for me, but I can give a general answer:

Roman soldiers did have to pay for their own armor, but they didn't have to buy it, if you see the distinction.

For most of Roman history, the equipment for Roman soldiers was provided for by the state. The state would buy the equipment from private producers, distribute it to the soldiers, and then withhold the cost of the equipment from the soldiers' pay over the period of a few years.

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Source:

P. Herz, Finances and Costs of the Roman Army (in: P. Erdkamp (ed.), A Companion to the Roman Army, Oxford 2007, p. 306-322)

Depending on the time period, yes. Up till the late Republic, the Roman army was a part-time militia and thus each man was responsible for purchasing their own equipment. So a wealthier man could afford a suit of mail armour (lorica hamata) while poorer soldiers would make do with a simple pectoral plate.

By the late Republic, Rome was wealthy enough to equip its armies. Plus a Roman manpower shortage (specifically a shortage of recruit-able men) led to Marius accepting volunteers who would be supplied by the state. From this point forward, armies were increasingly filled with volunteers from the landless segment who would look to their generals to reward them with land after a successful campaign.

Under Augustus, the professional army was formalized and everyone was supplied with equipment by the state.

Edit: Missed question about lorica segmentata. This was only in the imperial period and seems to have only been issued to the western legions. For some reason it was discontinued by the dominate, I read somewhere that the it may be because the fittings were easy to break.