Too many things influence the FOH house sound. What kind of speakers they are using, are they using any processing like compression, EQ or limiting? Keep in mind your sound is also being mixed to fit in with all the other musicians and that will change your sound completely.
1 - Use "Electric Guitar" mode as it was designed for use with multi-effects processors, PODs and pedal boards. However, and this is not meant to confuse you, you may like the high fidelity sound of "PA/Reference Mode." You are free to use it. Yes, the Flat Response, Full Range characteristic of the "PA Mode" might work better with certain models.
2 - A mic will not give you an accurate representation of the sound you hear coming from the L3T. Your talking about capturing signal from a speaker with 3 different components that reproduce in different parts of the spectrum. That's why you mentioned it sounds different than using the modes. Aiming the mic would be a challenge, but it may be worth experimenting with if you have the time. First choose "electric guitar" mode (or PA mode, if you prefer) then experiment like you would in the studio. No rules, though I don't recommend it.
3 - "Electric Guitar" mode is designed to have a different voicing. It emulates the frequency response of a 2x12, meaning its EQ curve is similar to that of a guitar amp and not a regular FOH speaker. Guitar amps have a characteristic sound due to the materials and speaker cones that are used. It's why guitars and effects sound unusual when you plug them into regular PA speakers. To more closely achieve the sound of a guitar through an amp, choose "Electric guitar" mode. Additionaly, in this mode, you take advantage of Virtual Tilt-Back, which optimizes the speaker’s tuning to achieve an upward tilt to its main axis and aim the sound towards the performer.
Other questions for you: Are you fedding the FOH the direct outs of your POD or from the "Mix Out" of your L3T?
Don't forget that if you only have one L3T, the "Mix Out" puts out a Mono Sum (L+R) of the input from the L6 link.
If you're using the outputs of the POD HD500x, always use the balanced XLR outputs, when you can. If you are using the 1/4IN outputs make sure it's set to "LINE" and you're using direct boxes to feed the front of house.