For the first couple of times, I'd suggest you sit down with a table in front of you. The table surface should ideally be higher than your waist in the sitting position. If you start to freak out, you can still be holding their body but have them climbing on the table.
As to getting them out, touch them on the side then pick up from the fattest part of their body.
Don't underestimate your instinct. When you have the bp out, you just know which part to support or move.
For checking the head, you can hold their chin up with your finger. My bp doesn't seem to find that invasive.