Looking to see if anyone has some advice for me...

My father-in-law has given me his 1987 F-150 5.0L 4x4 Lariat (4 on the floor, short bed) recently and I've driven it around a few times since receiving it. He had a 6inch rough country lift put on it as well as changed the stock tires to 35" tires (new rims as well but that is not the issue). In doing so I believe it is putting tremendous strain on 1st and 2nd gear. 3rd gear feels pretty average and 4th has no strain feeling what-so-ever. The strain I am feeling is that to get to any reasonable speed in 1st or 2nd gear my RPM's seem higher than they should be. Also, in those gears if I go to coast the truck has a sharp hauling feel. As in, the gearing is slowing it down even at a slow pace (5-15mph in 1st and 20-35 mph in 2nd). This sharp hauling is more of a nuisance than a problem I believe but I would like the truck to run as smoothly as possible. Over time that will damage the gear box, so I want to have it geared properly to handle the tires. The issue is so bad to the point that 1st gear is hardly usable and if it is, it must be shifted almost immediately into 2nd gear as the RPM's allow.

My questions are:

1) Do I need to replace my gear box to a 5 speed? If so, which one is recommended.
2) Do I need to re-gear my front and back differentials to accommodate for "non-stock" rotations per RPM? If so, what ratio is best?
3) Is there something I am not thinking about that needs to be looked at, changed, or upgraded to have the smoothest possible ride in all gears.

Let me know what y'all think if you've ever encountered a truck that was lifted too much for the stock gearing.

I am not sure which gear box is in it or the current gear ration or the axel type.