Thank you for contacting Ask Extension about your birch tree. We can provide some general information about pruning the tree; however, for a tree of this size, we do suggest contacting a professional arborist.

Regarding when to prune birch, the best time is when the tree goes dormant after the leaves have dropped. Avoid pruning birch between late April to mid-July since fresh wounds during this period can attract the bronze birch borer. 
For more information, see
http://pnwhandbooks.stage.extension.oregonstate.edu/insect/hort/landscape/hosts-pests-landscape-plants/birch-betula-bronze-birch-borer

If the tree has not recovered its previous form by now, it is not likely to straighten. Birch is not very tolerant of pruning, but if done carefully it can encourage and direct new growth in order to give the tree a more upright appearance. 

From your photos, I am not able to determine how far back you should trim the branches over your roof. It looks like there may be some very long limbs that extend over the roof. It may be necessary to remove these back to their origination point on a larger limb or the trunk. This type of cut is normally done just outside the branch collar so as to make the smallest wound. Frequently, the branch collar on birch limbs is hard to discern; however, you can often look for the ridge of bark between the trunk and a major limb to determine where the collar should be. 

In the first photo, there appears to be significant bark damage on the middle trunk and often this may involve some rot. Again, it would be best to have a professional inspect the tree for safety as well as aesthetics. 

Here is a link to the fundamentals of tree pruning. This information applies to smaller cuts and shrubs as well.
https://extension.oregonstate.edu/crop-production/fruit-trees/tree-pruning-basics

I hope this information is helpful. Feel free to contact us again if you have additional questions.