Just wondering if you need to kneel Bay of Ice to gain the 1 initiative on the card or if the card automatically provides 1 initiative to the owner who can then kneel to draw a card if they win initiative?

Does this also apply to the other cards that provide initiative i.e. the Clansman card that provides 2 initiative from Lions of the Rock

Passive bonus, no kneel required.

Yep, initiative and also income (i. e. +1G), don't require any kneeling.

Was there any specific reason you thought they needed to kneel?

As an additional consideration, a card provides its initiative or income bonus even if it is already knelt.

So, for that Clansman who provides an initiative bonus, for example, not only do you not need to kneel him in order to get the bonus, if he is knelt for some reason before plots are revealed, you still get the initiative bonus.

I thought they needed kneeling because you need to kneel cards that provide influence, therefore when there's cards that have gold and influence Eddard's Chambers etc I assumed you had to pick one

Ahhh, gotcha. Influence is a little weird as a resource. It doesn't behave like Initiative or Income as it really does nothing on it's own. It's not that for influence to work that it needs to be knelt, it's that certain cards require you to kneel other cards with influence to pay for their cost.

For example, a card that said "If you control 3 locations that provide influence, draw 1 additional card during the draw step." would actually allow you to draw that additional card without kneeling the influence providing locations. (That card does not exist, just an example)

Also, the influence and income on cards like the chambers are not mutually exclusive. You can absolutely get both in a turn. Unless there is effect to the contrary, you will always get the 1 income on a chamber. In addition, you will get the 1 influence, though as influence is almost always knelt to pay for costs you will usually only get the 1 and no more.