It is not likely a "handyman" would be able to do so, and it is not unreasonable to want this done if you are the homeowner suffering the loss. 

That $1,000 offer the tenant made is not an unreasonable offer to start.

Or is that an unreasonable thing to expect out of an agent?

This is an unreasonable complaint regarding businesses.

Is there a reason why all of them mentioned this but the above users claim its unreasonable?

This is an unreasonable complaint regarding businesses.

Some owners want to do some unreasonable and 'shady' things like you mentioned regarding the Tenant's security deposit and some others may be too far the other way... for example wanting to stop by every other week to check to see if the Tenant is okay... all very silly.

Every place I've stayed in past 2 year had entirely unreasonable ideas about how the place should be left.

Don’t want to be unreasonable demanding 3k two months ahead of time but also want money up front to take the property off the market.  

Unless the lease specifically says you can't and assuming the late fees are not unreasonable, you should be OK.3.