Any idea if there are different mortar mixes and how to match it to the brick?

Take the extra step in proving you're a good match and you won't fall out of escrow.  

Since this would be our first property we'd ideally like to purchase a home that would only need cosmetic fixes, so that we can keep it fully occupied with tenants.

Money Partner agreement seems ideal or as previously mentioned above, do it yourself and get the experience and work with people you trust. - If you can't find private money, while it's not the best option, you can try HMLs where I've seen they cover 95% purchase price and 100% rehab costs.

Ideally I would want to stay away from any additional encumbrances tied up in the property in addition to mortgages.

How do I deal with the agent?

. $50,000-$75,000 is ideal in my market!  

You may be putting this in an upstairs section so it wouldn't be ideal, but something that may help.

I would prefer to rehab and rent out so that sounds ideal.

Ideally, you should have asked for this at the time when you signed the agreement of sale.