Does Tenant ever pay common fees and taxes?

Theoretically, you always pay common fees and taxes, but not directly - they are baked into the rental price. The unit/building owner must pay taxes/common fees - after all, they are the owner and are ultimately responsible for all carrying costs. The owner might try to break it out separately to bait you with a seemingly low rent and then gouge you, but it's definitely not the norm. Is the owner having you write multiple checks per month?

I have rented lots of apartments in this city and never heard of a landlord trying that b.s. Should all be part of the rent. Sounds like a bait and switch just to get you to look at the place.

If not disclosed up front, this is obviously not right. In a coop sublet, the Board may request that the renter pay the maintenance directly to the corporation if there are issues with the shareholder (as in the shareholder has been late a couple of times recently or for whatever reason this is a pre-condition to the Board allowing the shareholder to sublet). I have not heard of this in a condo, but again, if it is made clear up front, there isn't necessarily anything wrong with it that jumps out at me.

Tenants usually pay their pro-rata share of common and real estate taxes if it is a commercial lease - not residential.

While in the US it may not be common for the tenant to pay common charges and tax, in much of continental Europe it is. I wonder if European buyers are now starting to try and rent out their units. Next you're going to see owners quoting weekly rental rates and telling tenants to put in their own kitchen appliances!