Happy birthday to "grantmeaname" who apparently just turned 20!
With regard to a HELOC as an emergency fund, that only works when the economy is strong, house prices are stable or rising, and banks are flush. In 2008-2009, pretty much everyone I know got their HELOCs frozen, reduced or eliminated. Banks were cutting risk exposure and a lot of folks had their HELOC emergency funds disappear just when they needed them. So while it's helpful to have a HELOC available to cover your needs in a personal emergency, it does not replace an emergency fund.
Not sure if you would benefit much by paying down the mortgage principal and refinancing. In your shoes, I would bulk up the savings and start on fully funding retirement ASAP. GF needs to do the same.
I think buying a new house is smart from the maintenance perspective. You may have to install some landscaping and fencing, but you do NOT need to paint for awhile. Interior decorating is a want, not a need. The two problems with new houses is they tend to be located on the outer edges of cities, away from work and shopping, and they often command a premium over older homes that may not be recovered in a quick resale. Buying any house is smart only if your employment is stable and you are sure you will live there for several years.