The Lex Papia Poppaea was a supplementary law modifying already existing marriage laws, which has been put into place some 25 years earlier by Augustus. The purpose of these laws was probably some combination of the following: Augustus trying to update the laws to reflect changing customs; trying to restore a sense of social order and morality in the upper class after the turmoil of the civil war; trying to raise the birthrate of citizens; using the "restoration of morals" as a tool to cement his own power.

As u/bigfridge224 said, our sources for these laws are fragmentary. We do know that women between the ages of 20 and 50, and men after the age of 25, were penalized for remaining unmarried or childless, committing adultery, or marrying a person of the wrong social rank.

"Unmarried persons, who are disabled by the Lex Julia [the initial form of the law, issued around 18 BCE] from taking inheritances and legacies, were formerly deemed capable of taking the benefit of a trust. And childless persons, who forfeit by the Lex Papia, on account of not having children, half of their inheritances and legacies, were formerly deemed capable of taking in full as beneficiaries of a trust" (Gaius, Institutiones 2.286)

So, the Lex Julia said that you had to be married if you wanted to inherit, and then the Lex Papia added that you also have to have kids.

As for rewards, freeborn women who had three children were no longer required to have a male guardian. Freedwomen (former slaves) benefited by gaining a certain degree of autonomy in the making of their wills if they had four children. Slaves, even after they were freed, usually were still dependent on their former owner, owing them a share of their profits and needing their permission for financial and legal transactions. The former owner would also inherit most of their property.

"The Lex Papia Poppaea afterwards exempted freedwomen from the tutelage of patrons, by prerogative of four children, and having established the rule that they could henceforth make testaments without the patron's authorization, it provided that a proportionate share of the freedwoman's property should be due to the patron, dependent on the number of her surviving children" (Ulpianus, Epitome 29.3-6).

Basically, if a former slave woman had 4 kids, she became legally able to make testaments, and when she died, some of her property would go to her kids instead of to her former owner.

The laws also restricted who was allowed to marry whom. It allowed anyone who was not a senator (or a senator's descendants) to marry freedwomen and have their children considered legitimate, but senators and their descendants (Senatorship was for life, and the Senate was not small) were not allowed to marry freedwomen, actresses, and children of actors/actresses. Other freeborn people were forbidden from marrying prostitutes, actresses, adulteresses, pimps, and slaves formerly owned by pimps.

The rate of infertility, infant death, and death in childbirth is really hard to nail down, but we know it was high. Lots of upper class, powerful Romans has wives or daughters die in childbirth, had no surviving children, or no children at all. Julius Caesar's daughter died in childbirth. Cicero's daughter died in childbirth with her second child, neither child survived. Augustus only had one child by any of his wives. Ovid, Seneca, and Pliny the younger each married three times, and only Ovid had a child (a daughter) grow to adulthood. Even the richest and best connected Romans were subject to these problems.

Upper-class Romans who wanted male heirs could adopt one (Julius Caesar adopted Augustus, and Augustus, after marrying his daughter off to three different people in an attempt to get her to produce a male heir, and eventually he ended up adopting his stepson, Tiberius). If you were especially important, you might get an exemption to the child-laws. Augustus exempted his wife, Livia, granting her the privilege of not having to have a guardian despite the fact that she had not born three children. The most common solution to the problem of not having kids with your current spouse was divorce. Divorce was very common in ancient Rome, and could be requested by either party (though only men had the right for automatic divorce because of adultery), and was generally accomplished without much fuss. Augustus, mentioned above, broke up more than one marriage in his quest for a male heir.

Women in the Classical World, by Fantham, Foley, et al. is an excellent source for this, and similar topics.

I can't speak with any authority on the medical element of your question, but let me have a crack at the first bit.

The problem with this law (and the Lex Julia de maritandis ordinibus of 18 BC, another piece of marriage legislation passed by the emperor Augustus) is that we don't have the full text of the law itself. All we can do is piece together bits from later legal commentaries on the laws, as well as the writings of Roman historians (Cassius Dio, Suetonius and Tacitus in particular) who wrote about why Augustus put the laws in place. If you're interested in looking at the primary evidence that Parkin and Pomeroy's "Roman Social History" has a collection of most of the relevant passages. Appendix 9 of Brunt's "Italian Manpower" also discusses the laws in detail.

From the sources we have it seems that the penalties for those without children or who didn't marry mostly involved restrictions on inheritance, with the Roman state taking a substantial cut. Childless or unmarried men may have been passed over for political promotions too, and might even have been banned from attending games in the circus and amphitheatre.

In terms of rewards, these mostly fell on women who had at least three children. These were then given the right to make wills and the freedom to engage in business without the oversight of a male guardian.

There is some disagreement among historians as to whether these laws were really aimed at increasing the population. Making inheritance easier and giving preferential treatment in political appointments would have had little impact on anyone below the senatorial elite. The stricter rules about the ages at which men and women could marry and the harsher punishments for adultery suggest that what Augustus was really doing was trying to strengthen the Roman family unit within the wealthiest sections of society. A central part of the imagery promoted around his rule was a return to traditional Roman values after the civil wars, and to my mind these marriage laws are a part of that. If you want to know more about the imagery around Augustus' rule then the standard work is still Zanker's "Power of Images in the Age of Augustus."