Sounds good, it looks like you have the setup to achieve this - have you tried creating a 'routine' in the Alexa app.
You could set a routine with a schedule to play Cats vs Dogs on the Firestick TV at 2pm for example but the big question is what is it going to cast to - Also Alexa is Glitchy, I just asked the fire remote to play cats vs dogs and it didn't switch over to the HDMI so I had to do that manually, but then it had located the film and I would need to click a few times to get it playing.
Everything to do with Alexa is Amazon prime related so Alexa devices are best at casting Amazon prime media that is without involving skills and adding extra layers of complexity; I think recently you can say play this on youtube like you would be able to do with a chromecast - I am seeing a lot of recent problems with people trying to cast Hulu in the last month, seems to be a disconnect with the skill so that the command is sticking on the hulu search screen and not playing what was asked for.
What should work though if you don't want to introduce logic and events and do this ad-hoc is loading the Alexa app - you have to click on the circle at the very bottom of the app in the middle - providing the app has the necessary permissions to access the mic on your phone you now have a mini echo dot in your hand, though having just tried this out it doesn't seem to work as well as the echo dots for controlling other devices even other Alexa enabled devices in the homegroup; Navigating to the Firestick device in the app there isn't really much to tweak.
I think you need to focus on getting it to work locally first and then work on controlling the TV remotely - Check CEC is turned on in your TV's settings first, this allows HDMI plugged in devices to control the TV, this might be called something else in the TV settings such as 'Control for HDMI' or something like that.
Also check the sources on your TV, if you have a firestick plugged in it will be on one of the HDMI slots and being a smart TV it will also have a source for Wireless display - If you pulled out a smartphone with a youtube app the casting button will pull up devices you can cast to, you should see Bobs Fire TV Stick, and then TV Make_30:56:b3:73:33:fc or something like that, so the second one is the Mac Adress of the TV, all modern smart TV's have a wireless display built in and you can cast directly to that - The Firestick or Google chrome is more like a virtual machine but can still be cast to like a wireless display - try casting something to both.
Alexa isn't very good at doing any of this really - I just tried this out, I first asked the Echo show to play youtube to Bob's Fire TV Stick, and that's what the device is named from when I set it up as default, a right mouthful 'Alexa play a youtube video of cats fighting dogs on Bob's Fire TV Stick' first go was a success but subsequent goes got either silence or a pause and 'did you mean the heater'.
Then I muted the echo show and moved onto the app and got the same result as you, the firestick is there in the devices but when I try to call on it to do anything it says that I need to set a skill up - I have had a brief look at the skills searching for 'Firestick' and don't see anything jumping out.
I'll have another go tomorrow and should be able to report back with something; I just use google Chromecast to work my TV's in general and have never had any problems - The routines mean you can type out some quite specific commands that would not be misinterpreted e.g 'play cats vs dogs on youtube on bob's TV' you would have to trial and error that though as google would sometimes load the TV and say sure here are some options so you would have to give further commands, or would play something completely unrelated.
Anyway you have a nice little problem here for the community and I am sure somebody more knowledgeable will be along to help - I will do some more field tests with my kit tomorrow as I've always got it at the back of my mind to explore my Alexa kit a bit more as I have lots of it, just use Google stuff as it has worked so far for me.
Sometimes it works well with Alexa but I have found there is a lot more effort to be put in, I was suprised when I got the Echo Shows to display the Blink Cameras on the screen easily - that was perhaps the easiest interaction with the amazon/alexa kit, otherwise it all has a lot of trial and error, the whole concept of skills works here and there - You would think that it would all be more streamlined especially with there own products -
Anyway will do a few more tests and report back, hopefully some other folk will chime in, would be nice to get this up and running - there is no reason that getting a movie to play on a firestick remotely with the Alexa app should be so cryptic as if it were a linux operating system.