Do all tenants make renters sign a lease?

A tenant IS a renter.

No.  There are many tenancies that exist without a contract and the law treats them as month to month renters.  That being said the choice to require a contract or not is up to the landlord, not the tenant.  

When you want to stay longer than the term of the lease your landlord may not ask you to sign a new contract and allow you to stay on a month to month basis where all terms of the tenancy remain the same as stated by the original lease.  But when renting month to month, the landlord (and the tenant) have the option to change any of the original terms with a full month's written notice.  So if the landlord wanted to raise the rent, they could with a months written notice and if the tenant refused the rent increase they would then need to give the landlord at least an entire month's notice that they intend to vacate.

I require all adults over 18 sign the lease for my rental properties.

The landlord has the tenant sign the lease not the other way around as you wrote it.  Yes most landlords require a lease. In most states you automatically go to a month to month.   Most landlords would require a new lease but some let you continue month to month. Exacty how they do it is 100% up to each individual landlord. 

Not always.  If you go beyond the lease with agreement of LL, it becomes month to month.

tenants can't make renters do anything...a landlord/owner/manager would require a lease.

Yes.

In which case, you don't get to live in the offered residence.