You did not say specifically why, unless its "looking old" was the sole reason, that you got rid of the old patio. Were you unhappy with the layout? ...The size? What function is the proposed patio to serve? Can we presume it would be an addition to the patio you now have?

First let me say - you have a lovely home and property! Since it is such a love place I would recommend hiring a landscape architect to really give your exterior and yard a well thought out plan. As for using brick for your patio - YES! Using a brick paver that matches or nearly matches the brick of your porch would seem to be a great way to tie the house and the landscaping together. Am totally in love with your stone fence as well! Good luck!

Privacy on your Patio is important. You also want peaceful and quiet. So my first observation is the AC compressor. Try placing some shrubbery adjacent to disguise it visually and soften sound when it runs I like bluestone, actually all natural surfaces. Leveling stone is more difficult both installing and furnature set upon them. Accordongly, I went with pavers. Your designer will help incorporate your wants within available space Things I consider include: Shade and sun Intimate and group dining Conversation Grilling Utilities Water feature Storage Possible fire pit Landscaping - we softened our planting beds but have a geometric patio Where a bistro table now rests was once a hot tub. Here’s a video of our patio. Hope this helps. Good luck. https://youtu.be/lOAwXzKfTxY

Why do you want a pergola?  Is it to provide shade?  Of course the direction your patio faces and the time of day come into play, but typically  you should cover at least three feet more than the area you wish to shade. The problem is that pergolas are usually the depth of the patio because that's aesthetically pleasing.  Be aware that if you plan to place furniture near the edge, those sitting there will not be in the shade.