Specifically, 74161 counters and 74377 registers, can I rely on them to be full of zeros on power on?
They do not have a defined state, and even if you observed some to power up in a specific state, there is no guarantee this will always happen -- on that sample, or any other. In addition, the specific state may depend on temperature; the previous (powered up) state (it may last a few seconds); the rate of rise of the power supply, and many other uncontrolled variables.
The data sheet for the device will indicate if there is a power-on defined state. If it does not specify a defined state then one cannot be assumed.
The 74161 has a clear (reset) input that can be held low during the power up transient time to guarantee all zeros, but otherwise the answer is no.
The 74377 has no reset or clear input and the data sheet does not specify a defined state so the answer is no.
Data sheets are easily found by a web search.
Not only they power-on in an unpredictable state, some 74xx devices sometimes power on in a self-inconsistent state that is not expected to happen in normal conditions.
e.g. a 7490 (a decimal counter) can power on with a pretty much non-decimal value. I have seen them start as 1100 or 1111.
No, the state at power-on is not guaranteed unless explicitly stated to be so.
And even if it was, one would have to be very naïve to blindly trust the kind of power-on reset circuit they'd put on there. Where they exist on ICs, they're garbage more often than not, especially in older chips, and you need to add external circuitry anyway to make the reset reliably occur under all possible conditions. Where such circuits are even characterized they'll typically require something like an unusual minimum supply voltage, or a minimum rate of rise of supply voltage (dv/dt) and won't handle blips or brown-outs in the supply voltage, for example not generating a reset under all reasonable conditions when a blip could corrupt the state of the chip. Or perhaps the length of the reset pulse is not guaranteed to be long enough to fully reset the chip.