I don't use a heat mat for brug seeds. I'd say soil that's at a comfortable room temp or a bit warmer is probably good but that's a guess. I have seeds from the same pod with vastly different germination rates. I transfer the seedlings to individual 4" pots when they have a couple sets of true leaves. I would give them a couple months before giving up and even then they might surprise you. I've had brugs sprout many months later in pots that I've recycled the soil in. I give the plants 14 hours minimum of light.

During this time of year I do use a heat mat for the seedlings. In the spring I do it out on the patio in indirect sunlight and vent them a little bit. I don't use a heat mat then and they sprout just fine. They will need a little humidity to get the seedcoats shed off.

Here is one of the little biodomes I was using last year. I had excellent results with it. It is just a little expensive to keep buying sponges if you have a lot of seeds. I use a numbering system on the foam insert to keep track of what cross is what and have a notebook that I write the corresponding cross/number in.
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This is what I put them in after they sprout. Little 3 oz dixie cups. Then up to the larger pots.
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This is what I do but you will find a lot of differences because there are lots of ways. Kind of like raising kids LOL!

~First I soak them in the little plastic dixie cups...you can use a permanent marker to lable them so they won't get mixed up.
~Then I peel the cork off and put them back in some water to soak with a little bit of peroxide added for a half hour or so.
~Next plant them in a sterile potting mix. You can sterilize your seed mix yourself or if you are using the biosponges you can soak them in boiled water with a few drops of bleach.
~Then plant them, and wait. Most should come right up in a week or two. I have had older seeds take up to 2 mos. or more. I give them light while they are germinating.

I forgot to say that the little dixie cups can be washed with a little bleach water and reused. The marker/writing comes off with a alcohol :)

Excellent information from the other gals and my bet is there will be different ways as others reply. I too love the bio domes for starting and will put them on the heat bed only certain times during the year. I would say certain varieties germinate faster than others with some taking up to six months to do so.

Is there a specific temperature we are trying to achieve to get germination results?
*60 degrees is what we use here.
Are some species harder to germinate than others?
**B. Arborea to me is the hardest and the longest wait period.

When do we pot them up after they germinate?
*** We wait until they have 3 sets of leaves before putting them into their own pots.

What is the longest I should wait for germination before I give up? (heating matt space issues)
**** If you start them and they are on the heat bed, no sprouting then I would toss them outside in the soil come spring when the threat of frost is gone. Mark the spot so you know where they are and see if they sprout outside.

How many hours of light a day do we give them?
***** 24/7 if you can afford the light, space and care