How do I use lemmas that are probably already proven, but including the proof myself is quicker than looking up a source?
You do need to make some effort to find in the literature a result you think is known, but after that what you propose is common and I feel is fine.  I do it sometimes.  In such a situation, I tend to say ``the following is probably known, but we include a proof for completeness.''
Even if you find the all your background lemmas in the literature, it might still make the reader's life easier if you include some or all of their proofs.  It is really annoying to have to request seven obscure papers by inter-library loan, and then figure out all the different authors' notation, just to fill in a few pages of proofs.