You should try posting in the Tropical Fruits section of this forum.
Yes, it can be amazing to inherit already well established old fruit trees. They usually produce impressive quality (and quantities) of fruit. Some types of fruit trees can take 15+ years before the fruit really reaches optimal flavor.

Papaya plant start fruit set at about 4 to 6 months of age
about the height of 4-5 ft. The fruit start ripening at about 4 to 6 month
after fruit set. Female and bisexual plant produce fruit, male plant produce
small fruit. The plant produce more fruit in it's young age and is best in health.
At the fourth year, about the height of 30-35 ft., the plant is more prone to disease
and difficult to harvest fruit because of height. It is wise to plant
additional plants every third year and cut old plants when new plants start
producing fruit. Pick light green/yellow papaya and store in room temperature
for soften in few days.

Broken papaya plant send several new branches but keep only
one branch if the plant is young, for older plant it is better to cut that
plant and start new one.

Papaya is heavy feeder, and need to fertilizer in 2-3 weeks
interval. Use mulch and keep the ground moist. They grow well in well drained
but moist soil. Roots are shallow and subject to rot if water logged.

^ That is if the plant is growing in the tropics. In central Florida, I would imagine the plant would not grow as fast.

In colder places the plants grow less fast, and fruits are mainly produced in the warmer seasons....anyway Papaya needs to grow, it can't stop growing for a long period....Anyway, Papaya is ready to be eaten when the it starts to become yellow outside, the inner has to be orange and soft....it can ripen also after the harvest.

when it is %25 yellow, you can pick it. %50 or more is a bit sweeter, but you start risking rot and animal predation.

if you dont have animal problems, and watch the fruit every day, pick when fully ripe (over %50 yellow)

trunks do not normally snap in 1/2 just from the weight of the fruit.

that said, i do top mine to try and keep them shorter and fruit lower to the ground.

you can buy dwarf seeds on ebay and university of Hawaii

(Waimanalo Low Bearing X-77 seeds)

$1 per pack (50), free shipping LOL...

http://www.ctahr.hawaii.edu/seed/seeds.asp

your current trees will start producing new leaves and stalks.

i keep a maximum of 3 stalks per plant. you will have more leaf area, so you can support a few more fruit per tree, but the same amount of roots, so, you should thin the flower buds / small fruits by 2/3 for 3 stalks

or by 1/2 for 2 stalks.

i even thin out flowers on single trunk plants by about %20, so the fruit are bigger and sweeter.

you can cut the top 2 inches off every month or so to restrict height

they like a lot of organics... fish emulsion, worm tea, compost tea...

they do not like wet-feet and need good drainage and mulch around the base of the tree.

better planted on a hill, or high point, raised bed etc... even if just 3 inches... roots will seek out water.

they are actually pretty drought tolerant,

though, grow a bit faster with moist soil and mulch