Let me take #1 very literally and answer that way. Can you use an external MIDI keyboard to use keys to control Montage to make noise? Sure. Can you use a pad controller as an alternative way to trigger sounds on Montage to make Montage respond with noises? Sure. Can you use either to make Montage into a sampler? No. A sampler has the function of recording and manipulating samples with the ability to edit sample playback loop points, do cross-fade, etc. Montage doesn't do this and adding an external "dumb" controller isn't going to make Montage become a sampler. Adding a computer with proper software will enable this.

Reading more into your question (not being as literal), I think you were more going for if you can use pads to play back noises and not really worried about how you can arrive there. Popular with pad controllers is playing back "one shot" samples. These are easy enough to load up into Montage - so you can arrive at a destination of assigning pads on a pad controller (external) to different samples without adding any cost as one-shot samples do not necessarily need loop points edited (they don't loop).

And #2 ...

A live set doesn't assign different parameters except for volume. Live set points to performances - and each performance can modify the parameters and how external controllers manipulate these parameters. Best would be to also edit your external controller to map to "default" CC numbers that preset performances use for assignable knobs/etc to avoid having to do half the programming. That said, each performance can change the CC number for MIDI receive of the ribbon, assignable switches (A.SW1, A.SW2), assignable knobs 1-8 (common), breath controller, and two expression pedals (foot controller 1 and 2). Therefore - potentially every user performance you program can conform to CC values of your external hardware and change the mapping of these controllers (assignable switches, knobs, expression, etc) on a per-performance basis. The assignment applies to all PARTs in the performance as receive CC programming is a common, not part-level parameter.

Once you map a controller to a CC number of your external controller - you can use motion control to make the controller affect a parameter. Such as making an assignable knob change cutoff. Or making an assignable switch change resonance. Or making the breath controller affect pitch. Etc.

Although you cannot reassign these CC numbers (or MIDI messages for pitch bend/aftertouch) - mod wheel, etc. can be used by the external controller if you assign a knob/slider/button on your external gear to trigger the standard CC numbers for these (MW= 1) or standard values for pitch bend/aftertouch (if your controller supports AT). As pitch bend/mod wheel/after touch are also available to participate in motion control which can target a pool of parameters.

Mod control is a matrix which maps controllers to parameters. Not all parameters are supported - but the list is extensive. See the documentation for a list:

https://usa.yamaha.com/files/download/other_assets/4/856354/montage_en_dl_d0.pdf

Page 171 "Controller Box Source" is a table of all of the controllers which can be used to target a parameter (aka destination). "Controller Box Destination" is a list of parameters that can be targeted. There are about 123 destinations. I won't count 124-251 since these are really part of the "motion control" infrastructure and not end-point parameters ("internal use"). I won't go into the nitty-gritty, but there are 2 "levels" of source-destination pairs (common or PART). Most apply to "PART" but a few handfuls apply to "common". This distinction isn't important now - but wanted to outline there are a few umbrella "classes" of parameters with different scopes.

Another note on motion control. There are two high-level limitations to be aware of. First, as already mentioned, not all parameters can be targeted. 123 is a lot - but there are many more than 123 parameters in a performance. Chances are you'll be fine with what Yamaha chose or can work around anything "missing". Second, only 16 destinations per PART can be assigned - so there's a hard-stop resource limitation. In practice you'll likely find this is not an issue. I think it's important to know it's not a bottomless pit that routes to everything - which is sometimes the notion or suggestion by enthusiastic demos/marketing info/etc. Do not let this diminish the reality that the routing matrix is still pretty large representing most parameters you'd wish for. The resource limitation depends much on your usage - so it's difficult to tell if 16 per PART will present any real limitation for your use. "YMMV".

There's different ways to skin this cat. You could also just use the standard pan/filter/etc. CC numbers instead of programming motion control if this short-list of parameters available to most standard MIDI keyboards fits the bill.

Since you can, in the context of a live set, add any performance you want - and the performance is where the programming goes for the motion control destinations - live set can indirectly be used. The direct method is using performances (user - or library if you wish to promote your user content to a more permanent library position) and live set will inherit whatever settings you make to the performance with exception of being able to override the performance volume.

1.- Is it possible to use external controllers connected to the Montage such as pad controllers (for sampling) and midi keyboards?

Yes. The Montage features both standard MIDI In, which will allow you use Pad Controllers and/or Keyboards with standard 5-pin MIDI Out, and it features an A/D Input so that you can run the audio Out from your external controllers through the Montage's Motion Control Engine. This not only allows you to take advantage of the Montage's powerful Effect processors but it's superior audio quality.

A pad Controller which can load custom samples (like the Yamaha DTX-M12) can be used not only as a separate sound source, but can provide input to the Montage Envelope Follower, it can drive the CLOCK so your LFOs, Effects, Arps, Motion Sequences, etc, can all be sync' to your controller. Also Montage can derive Tempo from audio... so it will follow the groove you set on the pad controller!

You can also modify sounds and parameters with your external controllers in ways that you may never have had access to before.

Please see this video for some ideas of how this can be used...
Mastering Montage Envelope Follower

2.- If possible. Can I assign different parameters to that controller (pad controllers / midi keyboards) for each live set?

Yes.
As you get to know the Montage you'll see that the LIVE SET is simply a list that is order in which you want to access those setups you will be using on stage.

Think of it as a Set list... as you create your USER PERFORMANCES you will configure your external devices as a PART/Zone within the Performance... you can pre-set all functions concerning your external devices and how they interact (or not) with the internal Montage Parts.