Here are some general guidelines on the use of science and technology in Amber and its surrounding environs.
Although Amber City seems Rennaisance-ish, and the countryside medieval-ish, the details of "what technology level does Amber appear to exist at" are much fuzzier, encompassing the spectrum from Roman Empire to the late-19th-century gaslight era, complicated by magical gadgetry.
This is, fundamentally, a fantasy setting, and whenever you use technology within Amber and the Golden Circle, you should keep in mind the fantasy underpinnings of theme. While we regard it as valid and interesting for people to attempt to introduce technology into Amber, part of the core recognition that needs to occur is that these efforts should ultimately fail. Stories of attempts to introduce technology into Amber are play-generating and fine, as long as they occur with the recognition that in the end, something will cause the technological progress to halt and roll back. Genuine working higher-technology solutions are inconsistent with theme.
Civil Engineering and Everyday Conveniences
Amber is not without certain conveniences of day-to-day living.
Simple machines and basic mechanical principles apply across shadow, and Amber thus has well-designed tools, and the knowledge of engineering principles, techniques of architecture and construction, etc. are well-known to those educated in such things. There is concrete for building, although the particular formula for this is unique to Amber. The knowledge of how to make reasonably pure glass exists, so it is available for clear windows.
There is running water, including, in most places, hot water. There is a reasonably effective sewer system, and a knowledge of the principles of hygiene and sanitation. In implementation, this is more closely akin to Roman Empire technology than anything else, although it is coupled with some ancient enchantments.
Street illumination in the city itself is provided by phosphorescent enchanted globes, rather than gaslight. A variety of other lighting methodologies are used indoors — primarily candles, torches, and oil lamps.
Medical technology is severely hampered by a relative lack of useful drugs. Some basic mechanical assistances invented in shadow exist in Amber in small quantities — syringes and needles, tubing for IVs, and so forth. Surgery is greatly hampered by the lack of effective anesthetics. Healing magics are a more common way to deal with severe issues, assuming that the victim can afford to pay for them.
Devices brought in from shadow might or might not work in another shadow. The more complex the technology, the less likely it is to work. Magic tends to be used as a replacement for technology, although magic, too, has its limitations, and limited cross-shadow applicability.
Certain abilities, including some advanced Pattern powers, allow technology to be adapted so it functions in Amber, but in general, these gifts are limited to single devices or limited use, and thus higher technology is not widespread in Amber.
Chemistry varies with shadow, and what is a hardy, stable material elsewhere may turn out not to be so in Amber (plastic and similar materials from Shadow Earth, for instance, or jeweler's rouge from Avalon).
In Amber, it is more appropriate to think of "alchemy" rather than chemistry. It is more art than science, and being wrong tends to be far more dangerous than it would if Amber chemistry were actually a consistent, predictable science.
The inherent stability of Amber itself makes it a poor place to attempt most mass-scale industrial processes.
That stability also means that very few substances explode in Amber, or at least, explode in a way that is useful. (Characters may have abilities that inadvertently end in disastrous explosions, but such explosions are negative consequences, never positive ones.) Until Corwin, no one had ever managed to bring in a stable explosive useful for military applications.
Limitations of RPG Gifts
Technology/science-oriented gifts also do not negate the need for other elements. Even if you possess a broad "create a device" gift, you cannot, for instance, simply assume a fire spell that powers a magipunk device, unless you have an appropriate gift that enables you to imbue something with fire.
The Golden Circle shadow of Begma has a steampunk esthetic, but it is more magipunk (blended magic/tech) or clockpunk (Rennaisnace) — steampunk itself is higher-tech than is appropriate in the setting. This is the most appropriate ways to introduce technology into the setting.
The key thing to Begman technology is that it fundamentally doesn't work — it's a mix of crazy technological invention bound together by magic, and in the end, using it is more dangerous and less effective than simply doing something via manual labor. Its primary use is really color, usually in a comic sense.
Yes, there are Begman gifts that allow the creation of working devices. Thematically, it should be kept in mind that these are weird, overly-complicated, unreliable inventions, even if a specific gift allows the device to be useful in a particular scene. Begman tokens are likely to be tightly policed by GMs for thematic appropriateness.
Note that fundamentally Begman devices are actually magic, and not technology per se. While Begmans, in-character, deny this utterly, this is objectively true (and part of Begma's backstory as a shadow). All the complicated gadgetry in the invention is really just a way of structuring the ritual magical power held by these devices.