hi, welcome to OB.
I think most people have had your thoughts at some point regarding hybridization
Even though it would be cool to cross a Cattleya with a Paphiopedilum, hybridization does not work that easy for orchids...
Tha basic rule (with very few exceptions!) is that you can only cross orchids that belong to the same tribe, e.g. Laelinae (Cattleya, Laelia, Sophronits...), Oncidinae (Oncidium, Odontoglossum, Psychopsis)... as mentioned, sometimes you have exceptions, that work in both directions: different tribes being able to be crossed, or same tribe not being able to be crossed...
Most orchids have the pollen (actually you speak about pollinia, as the pollen is packed and not in powder as in most flowers) in the tip of a structure called columna... Paphiopedilum and other slippers are a bit different because their "pollinia" are sticky and situated at both sides below the staminode, as explained by Glen above.
now, back to hybridization, I assume you do not know very well what have to be done in order to raise an orchid from seeds... this is as well not so easy (but not impossible!) and you should think whether all the work is worth for you: polination... wait 6 to 15 months to have the capsule ready to collect the seeds; send the seeds to a lab in order to have them sterile conditions (you can do this yourself if you have the equipments and knowledge!), wait for about 12 months until you can get the new plants out of the flasks and sent to you; wait for about 3 to 6 years until the plants that survived all this year produce the first flower...