Can activated abilities that tap be used more than once a turn?
like for example if I have Marwyn, the Nurturer and she is 3 power. I tap it to bring in Thorn Lieutenant. I know marwyn goes to 4 power due to ETB. and has 1 floating mana. can I use Stony Strength untap marwyn have her at 5 power tap again and add 5 mana to the pool for a total of 6 mana?
Another example would be incubation druid would I be able to tap him and use stony strength and untap and tap again to get 3 mana?
and for it's tapped ability if I had Incubation Druid out tapped and adapted it, could I just adapt it again if Biomancer's Familiar was tapped?
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If you can untap a creature with an ability with T in the cost, you can re-pay that cost. All that's required to pay the cost T is that the creature is untapped and doesn't have summoning sickness.
You can activate abilities that result in adapting even if the creature in question has +1/+1 counters, but in cases other than the use of Biomancer's Familiar, the adapt keyword action will do nothing. The Familiar reduces all activated ability costs, not just those that result in adapt.
An activated ability with T in its cost (such as Marwyn's second ability) can be activated only if the permanent with it is untapped and, if it's a creature, was under its controller's control since their most recent turn began. If such an ability is activated in a given turn and then that permanent becomes untapped, that ability can be activated again in the same turn upon tapping that permanent again (and paying the rest of the ability's cost). An activated ability that limits how often it can be activated in a turn will generally have text setting forth that restriction (e.g., "Activate this ability only once each turn") (C.R. 602.1b).
Shalai's and Elvish Chancellor's last abilities are activated abilities of creatures, so if Shalai or Elvish Chancellor is a creature "you control", Biomancer's Familiar's cost reduction can apply to the corresponding ability.
Biomancer's Familiar's second ability applies to the targeted creature the next time that creature adapts "this turn". For Incubation Druid, for example, it means that the next time Incubation Druid adapts due to its last ability "this turn", you put three +1/+1 counters on it regardless of how many +1/+1 counters it previously had.
EDIT: Correction after I saw comment 2.
EDIT(Feb. 13, 2019; Jan. 6, 2021): Further correctness edit.
Can activated abilities that tap be used more than once a turn?
like for example if I have Marwyn, the Nurturer and she is 3 power. I tap it to bring in Thorn Lieutenant. I know marwyn goes to 4 power due to ETB. and has 1 floating mana. can I use Stony Strength untap marwyn have her at 5 power tap again and add 5 mana to the pool for a total of 6 mana?
Another example would be incubation druid would I be able to tap him and use stony strength and untap and tap again to get 3 mana?
For Biomancer's familiar it's ability reduces mana sinks cost like? Shalai, voice of plenty and Elvish Clancaller so those would drop to 4 mana to activate instead of 6?
and for it's tapped ability if I had Incubation Druid out tapped and adapted it, could I just adapt it again if Biomancer's Familiar was tapped?
-Stay Frosty
You can activate abilities that result in adapting even if the creature in question has +1/+1 counters, but in cases other than the use of Biomancer's Familiar, the adapt keyword action will do nothing. The Familiar reduces all activated ability costs, not just those that result in adapt.
I̟̥͍̠ͅn̩͉̣͍̬͚ͅ ̬̬͖t̯̹̞̺͖͓̯̤h̘͍̬e͙̯͈̖̼̮ ̭̬f̺̲̲̪i͙͉̟̩̰r̪̝͚͈̝̥͍̝̲s̼̻͇̘̳͔ͅt̲̺̳̗̜̪̙ ̳̺̥̻͚̗ͅm̜̜̟̰͈͓͎͇o̝̖̮̝͇m̯̻̞̼̫̗͓̤e̩̯̬̮̩n͎̱̪̲̹͖t͇̖s̰̮ͅ,̤̲͙̻̭̻̯̹̰ ̖t̫̙̺̯͖͚̯ͅh͙̯̦̳̗̰̟e͖̪͉̼̯ ̪͕g̞̣͔a̗̦t̬̬͓͙̫̖̭̻e̩̻̯ ̜̖̦̖̤̭͙̬t̞̹̥̪͎͉ͅo͕͚͍͇̲͇͓̺ ̭̬͙͈̣̻t͈͍͙͓̫̖͙̩h̪̬̖̙e̗͈ ̗̬̟̞̺̤͉̯ͅa̦̯͚̙̜̮f͉͙̲̣̞̼t̪̤̞̣͚e̲͉̳̥r͇̪̙͚͓l̥̞̞͎̹̯̹ͅi͓̬f̮̥̬̞͈ͅe͎ ̟̩̤̳̠̯̩̯o̮̘̲p̟͚̣̞͉͓e͍̩̣n͔̼͕͚̜e̬̱d̼̘͎̖̹͍̮̠,͖̺̭̱̮ ̣̲͖̬̪̭̥a̪͚n̟̲̝̤̤̞̗d̘̱̗͇̮͕̳͕͔ ͖̞͉͎t̹̙͎h̰̱͉̗e̪̞̱̝̹̩ͅ ̠̱̩̭̦p̯̙e͓o̳͚̰̯̺̱̰͔̘p̬͎̱̣̼̩͇l̗̟̖͚̠e̱͉͔̱̦̬̟̙ ̖͚̪͔̼̦w̺̖̤̱e͖̗̻̦͓̖̘̜r̭̥e͔̹̫̱͕̦̰͕ ̗͔̠p̠̗͍͍̱̳̠r̰͔͎̰o͉̥͓̰͚̥s̟͚̹̱͔̣t͉̙̳̖͖̪̮r̥̘̥͙̹a͉̟̫̟̳̠̟̭t͈̜̰͈͎e̞̣̭̲̬ ͚̗̯̟͙i͍͖̰̘̦͖͉ṇ̮̻̯̦̲̩͍ ̦̮͚̫̤t͉͖̫͕ͅͅh͙̮̻̘̣̮̼e͕̺ ͙l͕̠͎̰̥i̲͓͉̲g̫̳̟͈͇̖h̠̦̖t͓̯͎̗ ̳̪̘̟̙̩̦o̫̲f̙͔̰̙̠ ̹̪̗͇̯t͖̼̼͉͖̬h̹͇̩e͚̖̺̤͉̹͕̪ ͚͓̭̝̺G͎̗̯̩o̫̯̮̟̮̳̘d̜̲͙̠-̩̳̯̲̗̜P̹̘̥͉̝h͍͈̗̖̝ͅa͍̗̮̼̗r̜̖͇̙̺a̭̺͔̞̳͈o̪̣͓̯̬͙̯̰̗h̖̦͈̥̯͔.͇̣̙̝
Shalai's and Elvish Chancellor's last abilities are activated abilities of creatures, so if Shalai or Elvish Chancellor is a creature "you control", Biomancer's Familiar's cost reduction can apply to the corresponding ability.
Biomancer's Familiar's second ability applies to the targeted creature the next time that creature adapts "this turn". For Incubation Druid, for example, it means that the next time Incubation Druid adapts due to its last ability "this turn", you put three +1/+1 counters on it regardless of how many +1/+1 counters it previously had.
EDIT: Correction after I saw comment 2.
EDIT(Feb. 13, 2019; Jan. 6, 2021): Further correctness edit.