1) Can I create a league, just like on the old NES? Technically, yes, but getting people to join would be difficult, you'd have to handle scheduling, make sure people submit games on time, set rules, likely hold a draft for players or teams, and before all of that, again, you'd need a playerbase. Because you haven't posted much here, players would likely be hesitant to join.
2) If yes, can we play games on mobile as well as laptops?
As far as I'm concerned, no. Using brudtopia/nestopia for online games is recommended.
3) If there are leagues, do we just log into the league to play our weekly games or is there more to it?
First you need to sign up for one. Keep in mind that most leagues have a wait list, so it will be a bit before you can join one. A good place to start, however, is TTL. After that, some leagues have a site, while others don't. Additionally, some leagues use save states to record game stats, while others, such as TTL, allow you to punch in the score on a google doc with a short recap of the game.
4) I have some technology challenged friends, how difficult is it to actually do this?
Setting up hosting is very difficult if you don't know what you're doing. I've tried to for a while and still have no idea how to. As for connecting to games, it's much easier. Simple copy-paste of the IP and you're in.
5) I would really like to know if these games can be played on mobile and laptops.
I believe there are NES emulators for mobile, however, as previously stated, it's extremely unlikely you'll be able to use them for online games. If you want to join a league and play some online games, you should download this. (If you're using a Mac, you can use wine to run the application.)
Hopefully these answer your questions, if you need any more information, or I misinterpreted a question, please let me know. - Tails