Position of dryer vent relative to gas meter and furnace intake
Good questions. There are a few requirements for a Dryer Vent, but none that impact your situation. See the International Mechanical Code's pertinent section here in the 2nd answer - https://dryerbox.com/building-code.htm

I'd suggest the new Dryer Vent hole approach and position it at about 1-ft. above either the top gas pipe or top furnace elbow. As, this location appears to be accessible with just a step stool and provides enough separation so heavy lint won't be introduced into your furnace filter.

This location seems to also keep the inside pipe below the top of the dryer and not crud-up the look of the laundry area and, quite importantly, keeps the vent pipe run as short as possible and therefore virtually self-cleaning. Please also use rigid duct pipe and zero flexible stuff for the best longevity and durability.

Concerning the old vent hole, which was horribly done and I completely disagree with a dryer vent that's less than 2-ft. above a horizontal surface. Maybe the doggy door could go there. Or, maybe there's a rear address number requirement that a plaque could remedy. Otherwise, pick up a couple of thin veneer brick tiles and get it looking right after some chiseling.

Don't forget to insulate the old vent hole before patching the inside. Also, to avoid lint getting to the furnace, consider a Louvered Dryer Vent Cap on the outside instead of a Hooded Dryer Vent Cap. The Louvered Cap blows straight out rather than downward.