I supply nice properties and respond quickly to any problems, in exchange for on time rent payments and keeping the property clean.

. - Sell & 1031: - I think I could sell right away for at least $225k for an $80k profit (not including closing or realtor fees).

Is this fully legal to exchange work for a reduced rent amount?

Only sell assets as strategic moves such as tax benefits, example 1031 etc. ( I personally do not have experience with this yet, but others do) Do not even think about this strategy, it will only add to the unknow and paralyzing fear. 

She could, however, use a 1031 exchange to leapfrog into investment properties with better cash flow.

The exchange rate risk could be huge. 

My mom purchase rental property in North Hollywood ( rent controlled area ) but she will leave in one unit.She buy this property cause was the last one affordable and few days left in 1031 exchange.Problem is that she is allergic to pets and in 6 unit apartment complex there are 9 dogs and one parrot bird, only one tenant doesn’t have animal out of 6 units.What are the options.

A "transaction" occurs when: Goods, services, or property are sold;Property is rented;Cash is exchanged for other cash;A contribution is made to a trust or escrow account;A loan is made or repaid; orCash is converted to a negotiable instrument, such as a check or a bond. 

I collected a security deposit and the tenant paid me 100% of the rent upfront in exchange for a 10% discount for his stay. 

Harder to raise rents even in nicely rehabbed units. 1031 Exchanged into a property in Arizona which I now do STR with.