so a quarterstaff with an ax strapped to it would have reach because its similar to a halberd, but a quarterstaff with a dagger strapped to it would not, because its similar to a spear?
No. Not long enough. Think about what it takes to make a reach attack. At an absolute minimum, you have to span a 5 foot gap. Then you have to hold the weapon, and you have to proceed at least halfway into the opponents square, and you can't assume you are at the very edge of your square. Hence a center to center distance is reasonable, 10 feet, plus length to hold it minus the length of your arms.
If something is not at least 10 feet long, it doesn't get reach, imo.
So, I would give an ax strapped to a quarterstaff an improvised weapon (1d4), but not reach.
Say a person with tavern brawler and only simple weapon proficiency picked up a halberd and said, " I want to use this as an improvised weapon." Would it be an improvised weapon (d4 damage) with reach in his hands?
Sure, but i suspect it would result in better average damage to just use it without proficiency as a halberd with reach (i.e. 1d8 right?) I guess at high levels your proficiency bonus may be worth a decrease in damage, but not at low levels.
do you mean lance? I'm not sure what a longspear is in 5e
I mean something much longer than a PHB spear. Lance is fine.
Sure, I mostly just interested to see in there is any consensus out there for these edge cases.
Also I'm considering house ruling that quarterstaves and spears are reach weapons when used two handed.
Nope, even worse, now you have a 6' shaft that you have to use about 3' of just to hold. And, since both arms have to hold it, now your backhand is even farther from your opponent and holding the weapon, so you have something like a 3' reach, not the needed 10 feet.