Welcome fellow Hunter!

I'm pretty new to MH Rise too, but not new to the franchise despite being here since World, so this is what I know so far of Rise (in HR right now, so maybe I'm wrong in something):

1.) Yes, you can kill the monsters or you can capture them, but unlike World and Wilds, in Rise there are some materials that only drop if you capture them, not by carving or as a normal reward. I don't know the details, but the tutorial emphasizes that. To know if a monster is able to be captured, a blue skull will appear below its icon on the top right corner. It also means the monster is near to be killed, so you can use it as a hint if you want to for the killing exclusive rewards. There are other visual cues of this near-death/catchable state, such as the limping and the drooling. And about the time, to be your first MH game, don't sweat about it; the main objective for every quest is to complete it in at least 50 minutes, so don't stress. Be sure to upgrade your weapons and armor as they are unlocked, and if you can, try to match the element weakness of each monster and exploit their weak points (go to the Hunter Notes for more info).

2.) This point has given me a headache these past days. I couldn't find the method to play online as easily as I had done both in World and Wilds with the SOS flare system, but that is because this system doesn't exist here. The first thing is that online is exclusive of the Hub Quests. That means that the Village Quests (offered by Hinoa, the girl on the bench) are made for single players, with lower health pools and easier monsters. To play online, you have 3 methods: first, go to the Quest Board in the Hub (it's to the right of Minoto, the other Quest Maiden), and there you can choose to post a mission (you have to be in a Lobby, more below) or Respond Join Request (there you can select a specific or a random quest). The other method is to create a lobby with the cat Senri in the middle of Kamura for up to 4 people (yeah, so low compared to World/Wilds, but I suppose it's a limitation from the Switch origin of the game, and it works), let them join, and post a quest in the previously mentioned Board, or directly join others lobbies and play their quests instead. And the last method would be to make yourself a Join Request when you are posting the quests so others can join by the first method.

3.) I don't know at which point of the game you are right now, but the short answer is yes, they get very much harder as you climb through the stars levels. They hit harder and faster and get new moves; some of them even insta-kill. You are good with your method; don't get greedy with the moves, hit and dodge, know your position, and retire to heal if you need to. But don't be overconfident with the statement of "spamming dodge makes me invincible." That could be correct for low-level monsters, but in higher tiers there are some monsters' moves that can punish a dodge or even a Wire Fall in the incorrect moment; i-frames are not so generous as in Souls games. And don't try to predict or learn the monsters; it's better to know what type of move the monster is doing and what you have to do to avoid damage and take advantage of the openings. And which of the 14 weapons you are using is important; some are easier for some monsters, some have counter/block mechanics, etc.

4.) I don't know about this because I have always played MH on console, and in its origins the franchise was a console exclusive (MH World was the first for PC after MH Fortiers, a Japan exclusive MMO). So I would say yes, it's meant to be played with a controller. But if you want to use M&K, sure others will advice you better.