How effective and safe were condoms at preventing pregnancy and STI's before the advent of modern, latex condoms?

I can only speak for the period prior to 1800 but, in short, yes, early modern condoms were probably effective in preventing sexually transmitted diseases, which was their original intended purpose. However, they do not appear to have been commonly thought of as a contraceptive in the pre-modern period (although they may have had contraceptive effects.)

The earliest accounts that clearly described condom use come from the early sixteenth century. There is some evidence that condoms and other barrier methods had been used before, but their use did not become well-documented until the outbreak of the syphilis epidemic in Europe at the end of the fifteenth century. Originally their use was thought of as a means to prevent disease, not pregnancy. That is, condoms protected men from disease, as some thought that women were responsible for the spread of venereal disease, but could not contract it themselves (perhaps due to the asymptomatic nature of a lot of STIs in women). The few records of early modern condom use make clear that men seemingly exclusively used condoms with prostitutes, which would have also made pregnancy prevention (from the male perspective) less of a priority, as there would be no binding legal or social obligation on their part if pregnancy did occur. Furthermore, medical authorities had often claimed that prostitutes were sterile to begin with because, in the view of William of Conches (ca. 1080-1154), frequent intercourse would make the womb "slippery like marble" and "unable to hold onto the sperm." Furthermore, diseases like syphilis left visible marks on the skin which were seen as incredibly shameful and could compromise a man's reputation and marriage prospects, whereas a prostitute had little to lose in terms of social status from an unplanned pregnancy. This is not to say that women did not use birth control or fail to see the utility of condoms as a contraceptive, but that most of the existing early modern sources only record men's reasoning behind condom use, which suggests a preoccupation with disease rather than pregnancy.

The earliest descriptions of condoms thus only refer to condoms as a method for preventing disease. Gabriello Fallopio--the same who named the Fallopian tubes--claimed that he had invented the condom to prevent the contraction of syphilis in a 1564 treatise on the disease, De Morbo Gallico. He described it as a lubricated piece of linen that only fit on the glans of the penis or, alternatively, inserted into the urethra. According to him, condoms did successfully prevent disease: "I tried the experiment on eleven hundred men, and I call immortal God to witness that not one of them was infected." Although Fallopio claimed that condoms did successfully prevent disease, however, he did not follow up with any of the women involved. Otherwise, he may have observed that condoms have an additional use as a contraceptive.

Preventing pregnancy thus probably wasn't the cultural purpose condoms were originally used for, but it probably was not long before people realized that they could also prevent pregnancy. In the eighteenth century, it seems people more often used animal skins and bladders and these "machines" were marketed as preventing both disease and pregnancy. It is hard to say how these methods would have compared in effectiveness to Fallopio's linen contraption. I would imagine that people probably experienced the same risks with condoms as they do with modern condoms in that they could break (in the case of animal skins) or come off during intercourse (in the case of cloth sheaths).

In any case, the paucity of sources describing them suggests that condom use was not especially common prior to the modern period. They were only used by a small minority of men who frequented brothels, such as the notorious rake Casanova, who described using condoms with prostitutes. These men were primarily concerned with preventing disease rather than pregnancy. Historians have found much more evidence to suggest that early modern people preferred natural contraceptive methods. A steep drop-off in birth-rates in the eighteenth century has been credited to widespread practice of birth control, but most probably in the form of coitus interruptus rather than in condom use.