I'm going answer these in weird order:
Would executing another king destabilze the region?
No, it does almost nothing, you remove that party and a new one rides out to take it's place, a few executions does nothing but give you -relations.
is executing lords worth it?
Executing lords is an all in strategy to quickly finish the game, but it's only good if you intend to kill the entire roster of lords, including any rebel clans that spawn. You will quickly rack up negative relations all over so you need be prepared to maintain your force via forced recruitment, bandits, and prisoners and of course minimal losses. It will be very very expensive to recruit any vassals or get married with negative relations (some recruitment may not be possible?) and I don't know how much it effect vassal loyalty. I don't how much it effect clan you raise for wanderers. Execution for me is an all in way to play a quick game where I don't do any siege or kingdom stuff, I just kill everyone, one faction at a time and when the dust settles the game is forced to give my clan all the fiefs as I'm the only one left, the it's GAME OVER you win.
does giving my vasal clan more fiefs means that they will have a bigger army
Yes, but it's not a fast or dramatic change, but they will tend to have fuller, higher leveled parties, recruit faster and have more influence.
However if you have a choice between this using a fief to create a new clan from ma wanderer, I would ALWAYS just create a new clan.
r does that mean that they will be more richer and thus have more members?
They can field 2-3 parties depending on their clan rank (and a perk that I think nobody has lol), having more wealth may help them maintain those parties but they won't make more then that and IMO the AI isn't really good at using wealth to create a marshal advantage.
so I can't create big armies.
You gotta carry the young faction. There's a couple policies (like royal guard) that give significant increase to you party size. What I do in new faction is make a full party of 200-300 HA (khan's guards + whatever else) and just zoom around (not in an army) stopping enemies in the field while I let my clan parties and early vassals just kind get their feet wet on their own. This gets me influence fast so I can get any need policies passed before I have people voting against me and the I can use each 500 influence to make new clan from a wanderer.
ow do I level up my engenieer skill and how do I level up my companion engenieer skill?
Engineer skill comes form building siege camp, building siege gear and destroying enemy siege gear with siege gear on the map. Make sure the person you set as engineer (or you) has fp invested into the skill.
1 Set the person as the engineer in you party 2 siege a fief a que up 4 trebs and then siege towers and the ram (in that order), each time a treb is finished pause and put it in reserve, when all 4 are ready set them all out and they start to destroy the enemy gear (good skill ups) while the other gear is build, let them keep going to destroy the walls (sadly no skill) the enemy will keep putting up gears that your trebs will destroy, continue until the walls are destroyed and then you have a choice: If you want to finish the siege do so (remove the ram though lol)! However if you want more engineer skill you can prolong the siege by removing 1 treb and building 1 extra gear at a time while the 3 trebs shoot down any gear the enemy makes, make all 4 ballista (one at a time!) and all 4 Orongor... magonels? Keep 3 trebs out though to kill the enemy gear quickly! Now when you've built everything you can either just do the siege (magonel is good in actual seige!) OR if you just want engineer skill, leave the camp and re-siege and build it all again!
Of course enemies will come and come the longer you do siege stuff, but that just saves you from having to hunt them down!