I've seen sheet music where you beam together a 1/8 note with a dot next to it and a 1/16 note at the end. Basically the 1/16 note has a second beam coming out but it doesn't extend. Can you do the same for a normal 1/8 note and a 1/16 note? Or do you just write them as two separate notes?
Yes, you can. Here's the most common example:
The alternate way of writing this is harder to read:
The other obvious examples are semiquaver-semiquaver-quaver, and its second cousin twice removed, quaver-semiquaver-semiquaver. I feel that they are less in the spirit of your question, though.
Most of the other examples are derived from these forms, with some of the notes replaced with rests.
Generally speaking, you should try and make the rhythms easy to read. If the best option is beaming a quaver and a semiquaver, do it. Just don't do this sort of thing:
This is much better: