I feel like your best bet is still using Laravel, I don't know the specifics on what lumen has or not but you're not losing anything by choosing laravel, especially if you're building a traditional web app. If you are building just the API I would totally go for lumen.
You don't have to use vue router, you can just use traditional routing if you don't mind page reloads. I like using both blade and vue components.
VueX? Nah, don't really need it unless your application is going to be huge. If it is going to be huge I would still doubt really needing it.
I wouldn't know the answer to your last question but since it's Lumen specific using Laravel will be your best bet.