Muleek (spelling varies) is a cluster of buildings along the modern Mulleek Rd. (Road names were only introduced in that area in the 1960s. In the 1840s your townland alone was sufficient to identify you and get a letter delivered).
Muleek is scattered across 3 townlands: Leggs, Tirigannon and Ballymagaghran. There’s nothing much there apart from a ruined church, a Community Centre and there was a National School there which your ancestors might have attended. It was later renamed St Davog’s.
Some school attendance records exist for that school, but I don’t know how far back they go. They might go back to the 1840s but that would be fairly rare. They are contained in SCH 291, SCH 1435 & SCH 1436 at PRONI. They are not on-line, so a personal visit is required to view them.
A townland is the basic land unit in Ireland. The whole country is divided into tens of thousands of them. Mulleek is on the Placenames NI site which has it (Mulleek) and explains it means “middle bay” and is in the civil parish of Belleek.
https://experience.arcgis.com/experience/9b31e0501b744154b4584b1dce1f859b
The PRONI e-catalaogue has several references to Muleek & Mulleek but most are 20th century and would not mention your family.
I checked Griffiths Valuation for the 3 townland that Muleek is in. No Muldoons listed in any of them. Very close to Muleek however is Lowerybane townland (previously Lowry Johnston). In 1832 there was a Charles Muldoon farming there, according to the tithe applotment records. Perhaps a connection?:
http://www.irishgenealogyhub.com/fermanagh/tithe-applotment-books/parish-of-belleek.php
Lowerybane is on the shore of the Lough and is about 1 mile east of Belleek village. The family had gone by the time of Griffiths Valuation in 1862 which obviously fits with your information. Lowerybane then was 248 acres of agricultural land with about 5 farms on it. It’s very much the same today with a few more houses. Agriculture and tourism would be the main employers in that area. The modern Boa Island Rd goes through Lowerybane.
In the 1901 census there were 5 homes in Lowerybane and a population of 23. Mostly farmers with 2 grocers (so there was a shop) and 1 railway worker (the railway from Enniskillen to Bundoran ran through the middle of the townland. Opened 1866).
http://www.census.nationalarchives.ie/pages/1901/Fermanagh/Castlecaldwell/Lowerybane/
That townland is in the RC parish of Carn. Unfortunately it has no records before 1851.
Muldoon is not a very common name in Fermanagh. In the 1901 census there were just 74. One fairly near to Muleek (a mile or so away) :
http://www.census.nationalarchives.ie/pages/1901/Fermanagh/Castlecaldwe…
You could try Enniskillen Library, local studies section for famine information. They may also have a copy of the Ordnance Survey memoirs of the parish c 1836 which would give a general description of it then. (The OS memoirs are not on-line so you need to either buy a copy or get someone to copy the relevant pages for Belleek parish). The library will surely have a copy. Contact details here:
https://www.librariesni.org.uk/libraries/county-fermanagh/enniskillen-library/
If Mary & Catherine Muldoon died in Scotland, their death certificates should record their parents names (if known to the informant). That might clarify the mother’s name including her maiden name. The records should be on the Scotlandspeople site. It should also contain Charles’ occupation. If he was a farmer (as opposed to a labourer) then that might provide a hint that he’s the man listed in the tithe applotment records.