Electrical – Tried to move the doorbell over a few feet but now it constantly rings like a school fire alarm

Apparently, the new doorbell is made to operate at 16 volts not 10. It seems to not work well under-voltaged.

I would install the new transformer and disconnect the wireless extender for a while until you determine if the new door bell is working properly. Just make sure you shut off the circuit breaker feeding the old transformer before replacing it. Double check to make sure power is shut off before replacing it.

Good luck!

the transformer is pretty warm to the touch....Not crazy hot but deff warm. Is this normal for a transformer?

A loaded transformer should be about as warm as your laptop charger. On the conceptual level, they do essentially same thing. (the transformer is less efficient, but the laptop charger puts much more power so the waste heat should be similar)

Or do I have to much power being distributed to it?

The only way of giving it too much power would be wiring a 110V transformer (yours) to a 220V circuit. Or wiring it backwards, but that would have blown up already.

the transformer makes a decent amount of noise.

That's bit concerning. A transformer should emit mains hum, but it should be very faint, impossible to hear from a distance. On a busy day, you should have to put your ear to it in order to notice the sound, easier to feel by hand. Unless something is acting as sound amplifier (like a guitar body), eg. being fixed to a drywall or lying on a desk.

I guess my overall concern is I understand the very (and I mean very) basics of a transformer and resistor. But are the above items normal? Should a transformer be silent and cool? Based on my set up is there any red flags?

The phenomena you're describing are fundamentally normal, but it's impossible to tell from your description if the intensity is normal as well. Cheaply made, loose transformer will make much more noise than a decent unit.

Generally, when dealing with a new circuit, the best course is to assemble and run it on a bench first. In stages. You should observe the transformer running at idle (without any load), with almost-idle load (videobell fully charged), with small load (the videobell charging through the resistor) and at full load (with the bell button depressed / the resistor hooked up directly). The sound and heat of the transformer change with load. The heat load of your resistor also change, depending if the button is pressed. You could try using the 8, 16 or 24V taps to see which one works best. Contra-intuitively, using higher voltage sometimes resuls in smaller losses when charging an electronic device. You could try to use 2 resistors in series (you've linked a 2-pack), especially with 16 or 24V. Even a cheapest multimeter would help greatly to ensure you get expected voltages. (Within a healthy margin, certainly. A doorbell transformer is not meant to be precise.)

The most important question would be: is the videobell charging? All of your issues could be easily explained if some of the output terminals are shorted, overloading the transformer. The unused terminal should be left unconnected.

Ring uses 30VA transformer on their examples, yours has only 10VA on 8 and 16V circuits and 20VA on 24V - but that's definitely not a problem here. With 25ohm resistor, the power always stays within the limits. On 8V it's merely 2 watts (you can assume that 1VA = 1W in this case).