If I use a USB-C -> USB-A adapter, can I expect the device to work as normal, just with type-A speeds? Or is there more to the USB-C standard that could cause the device to not work properly?
An adapter should handle any differences or adjustments between the connector types, provided it is an adapter appropriate for the job.
In other words, if the adapter says it'll work for you needs (power/charging vs connecting the device to a computer) I'd trust it's claim and report issues as warranty problems.
Depends a lot on the cable, just like there are some cables that are USB 2.0 but have just 2 internal wires instead of 4 so it works just for power, but not for data (so called "charging cables", usually chinese knockoffs)
The capabilities of USB Type-C cables depends on (1) USB modes supported, (2) Amount of power supported for charging and (3) Alternate modes supported by the cable. There are cables which support only USB 2.0 with up to 480 Mbps data rate. The cables which support USB 3.0/USB 3.1 Gen 1 can handle up to 5 Gbps data rate at full duplex and are marked as super-speed cable. The cables which support USB 3.1 Gen 2 handle up to 10 Gbps data rate at full duplex and are marked as super-speed+. The charging capacity of USB Type-C cables may go up to 5A (100W), but 3A (i.e. up to 60W) cables are commonly available. Support for alternate modes requires extra connections for optional pins as well - like additional super-speed lanes, and 'side-band use' pins. The support for alternate modes usually adds an extra bandwidth requirement to the cable - in addition to the bandwidth already needed for USB data transfer. There are USB-IF certification programs available for cables.
USB Type-C is a 24-pin plug so there's a lot more potential for making cheap knockoffs just for charging, or slower bandwidth. For USB-A I'd say you should be good to go, but doesn't hurt to buy quality cables. USB Type-C specification defines two differential pairs for USB 2.0 data bus, so it should be compatible (if the wires actually do exist in the cable, that is). That said, I rather doubt they would make a charging-only adapter, so yeah.
If I use a USB-C -> USB-A adapter, can I expect the device to work as normal,
Yes, but... there are "legacy cables", and "legacy adapter cables". They all are cables.
just with type-A speeds?
-no, since there is no "just Type-A speed", the Type-A USB3.1 connector is perfectly capable of SuperSpeed+ (Gen2) 10Gbps speeds, at least those who are certified as such.
So yes, if you use an appropriate "Legacy
Adapter Cable", your device is expected to work as normal. The Type-C standard specifies several variants of Type-C (plug) to Type-A (plug) cables, one group for USB 2.0 only speeds, and the other group for USB 3.1 speeds. The USB 3.1 cable assembly must have all good-quality wires to support the Gen2 10Gbps speed. Therefore your device will not suffer at all.
The legacy Type-C-to-Type-A cables have reduced number of data wires, only USB 2.0 D+ and D- wires. The cable can be identified by white plasic (not blue) inside the Type-A plug. This cable will work for your device too, but at the USB 2.0 rate.
The Type-C plug of a legacy cable must have its CC pin pulled up to VBUS with 56k resistor. This value indicates to your device that this is a standard USB port, so the device will not to try charging itself with excessive current that the standard port cannot handle. Cables with 22k and 10k pullups are illegal and might cause damage to USB host computer, there were few scandals regarding "dangerous Type-C cables"
The "plug-plug" cables should be distinguished from "Type-C plug -> legacy receptacle, aka adapter assemblies". The standard defines only two types of legacy adapters, micro-B USB2.0 receptacle to Type-C plug, and full-featured USB 3.1 Type-A receptacle to Type-C plug. In this case the C-plug must have 5.1k pull-down on CC pin, indicating that the Type-C port should reverse its role to HOST mode. Therefore, if your device is capable of Dual-Role-Data mode (formerly OTG), it should provide VBUS, and you should be able to connect a mouse or keyboard or a self-powered hub.
In total, the Type-C connector specifications v1.2 defines 15 (fifteen!) types of cables (there used to be 17). Welcome to the future, of USB cable confusions.