I live in the country and want to extend my wifi to a building some distance away. I've used Mikrotik routerOS boxes for my firewall, so I thought I would try building a wifi extender using Mikrotik.

I have wifi 'n' at the main house. I need to connect to the n network on one side, and want to offer n (or other) on the other wise.

Can someone tell me which Mikrotik device I should use for this? (I found videos saying to connecting devices back to back, but found another comment that a single device could be but bridge and AP_

Can someone tell me which Mikrotik device I should use for this? (I found videos saying to connecting devices back to back, but found another comment that a single device could be but bridge and AP_

"Cheapest possible" is not the best thing to be looking for. You can use almost any MikroTik WiFi device as an extender down to the most inexpensive 2.4GHz models. However, any such extender device that doubles as the main WiFi access point in your home will slow down your wireless performance since the wireless interface is asked to do double duty - both being an extender and serving wireless clients. The "Audience" solves this issue by having two 5GHz radios - one for uplink/downlink to another Audience, and the other for normal 5GHz client communication. You don't necessarily have to purchase Audience units for this if you don't feel you need 5GHz for clients in your case, but the concept of using a separate radio for uplink to the upstream radio and downlink to the downstream radio makes a great deal of sense.

Can I use the 2.4Ghz link to my house, and 5GHz out the other side? (So I don't need the $250 audience model)?

Is there a model with higher output radios? I see the HAP AC3 claims high power.

Also, is there a guide to setting up this kind of dual extended + AP mode?

You did not mention what that "some distance" was. For short ranges you could indeed try the omnidirectional devices for that link, it's far from ideal, but it could work.
If that distance is more than 20 meter (outdoor), you better go for a directional device for that link.

One low cost solution is using 2 SXTsq (there are several models, the SXTsq 5 ac is the best in the SXTsq range).
Setup is very easy : set one SXTsq WLAN in "bridge" mode (level 3 license cannot be "ap-bridge") , and the other SXTsq WLAN as "station bridge" mode.
Add all interfaces to the bridge in RouterOS. (ether1 and WLAN1) in both SXTsq. You have now a bridged/switched network with 2 ethernet ports.
There is no need for WDS to distribute the LAN over those ethernet ports, the "bridge"/"station bridge" WLAN modes will make them part of one level2 LAN.
On those ethernet ports connect you local access points. They will work as if they are ethernet connected via a switch.
To optimize performance use different channels in the SXTsq versus the local AP's. You can even set the "nv2" protocol for the SXTsq connection.

Besides this low cost, there are many more possibilities .... depends on what you need.

I have a Groover A-2Hn with a 2.4Ghz YAGI antenna pointing the signal in the right direction (from the house).

So I'm hoping for some routerOS device with 2.4 + 5 GHz radios (one to the home link, the other as the local AP). I imagine external antenna's we be preferable as that should have higher gain.

It would be nice if the 2.4Ghz antenna was directional so I could point it towards the house.