Whether you draw depends on two things as I see it:

a) the number of colours in your deck.
b) what you know about the opponents deck.

Anyway, if you are 2 colour, you PLAY. If you are 3, then in invasion I would probably decide based on my opponent. FOr mirror colours I would like to draw, for opposing colours the game can easily be a protecttion grizzly race, so you'll need the turn. Above 3 colours (yikes) , you draw.

As for mulligan, that depends a lot on the deck you design. You must consider

a) do you have enough mana
b) do you have enough spells
c) do you have the right colours of mana
d) do you have any cantrips, land search cards that you can play.

In sealed I really dislike parising, but since there will be a lot of 3 colour around, this is an important question. Basically most multi decks should have 2-5 cards that help bring the mana distribution together. If you have and can play a harrow 3rd turn for example, your golden. Basically though, I won't paris unless I look at my hand and know that I will not be able to develop (no land or all wrong colour) unless I draw well. The choice that kills, however, is the 2 land hand. Maybe just right, maybe a disaster waiting. Here it usually depends if Im playing, drawing.