Do Americans have a council tax system like in the UK where you have to pay for rubbish collections and things like street lighting?

Yes we have to pay for all these things (nothing is free).   But how we pay for it and who we pay varies by location.

Where I live trash collection is a service provided by a private business.   I can pay for it if I want curbside trash/recycling/compost or I can opt out and deal with my trash on my own.   In the large urban city 15 miles away, paying the city for garbage/recycling/compost is a requirement.  You do get your choice of small, medium, large though.

Property tax funds a lot of the local city/county services like street lights (and schools, parks, police, etc.).  Property tax is paid by people who own property.   Tenants indirectly pay property tax because it's built into what landlords charge for rent.

I pay a monthly bill to the city I live in for water, sewer and trash/recycling pickup. Things like street lighting, etc are paid semiannualy via my property taxes.

In some places, trash is a municipal responsibility, others it is contracted to private companies, and in unincorporated area, I wouldn't be surprised if trash is still just dumped anywhere.  Street lighting is usually handled by the government, it isn't called a council and nobody votes directly for those people who are in charge of street lighting (I don't know how it works in uk, so can't say yes or no)

We never know what our tax money is spent on