Hello Coleene,
Regarding the time frame for the egg appearance time  from a virgin queen
which
traveled with a swarm.  Quick, about 4-5 days.  The bees are VERY anxious for
her to lay tons of eggs, so they "push her out" on her mating flight just a
day or two after
selection of a new homesite, and she will lay the first few eggs about 48
hours after mating.

Regarding how long before the new virgin queen in parent hive will start
laying.
Starting from the date of the swarm, the first sign of eggs might be as long
as 14 days, but might be a few days less. Dependent on the age of the queen
cell when the
swarm departed (could be 1 day or 5 days to virgin emergence), then the virgin
will rarely go out to mate until she is 6 days old from emergence, and she
doesn't lay eggs for 2 days after being mated.

1:1 sugar syrup is the CORRECT dilution - do NOT make it heavier!  Heavy
syrup is strictly for winter storage; but thin syrup is artificial nectar
which stimulates
egg laying and comb building.  FEED CONTINUOUSLY until September.

Hope I have helped.

George Imirie