Tax on Gift and Property pass down

Hi,

Questions / Clarifications:

1. For the annual gift tax exclusion ($19,000), person-A simply writes a check of $19K to person-B.  No need to report anything to both their IRS filings, correct?
2. (Property Pass Down) The elders passed away and handed their (20 year) property to their heirs, 
   1. (Scenario 1) if their heirs decide to sell at current market, what is the capital gain based on?
   2. (Scenario 2) if their heirs decide to keep and rent it out, are they any tax implications?
   3. (Property Tax) Will it jump to current appreciation rate?   Or, inherit the elder's property tax rate?

Thank you!