The car only works in reverse, and/or won't accelerate in third gear. I need to start in first and ease into third.. seems to skip over second gear. At each stop car needs to first accelerate in first. Whats the problem?

I would say that with the grey smoke and all, your piston/s are skipping.
This means that the cylinders are worn unevenly from top to bottom.
The pistons then slightly bounce or skip in the cylinders.
Time for a rebore perhaps?

Check the transmission computer

"Generally" in manual transmissions, first and reverse gear are on the same gear cluster, thus if one works both do.
However, the remaining gears are on what 'used' to be referred to as the 'cluster gear' and this 'cluster' could fail, still leaving you with first and reverse gears.
Hope this helps.

no gears indicates that the torque converter clutch servo/solenoid s faulty
probably a disconnected wire from your working with the others

I suggest you take it back to whoever did your transmission.

this problem is a common failure associated with your shift solenoinds inside your tranny. of course a shop is not gonna tell you this because by the time you wanna get it repaired it is usually too late and have already cause damage to your clutch pack and internal parts, thats the main reason why a shop wont mess with just replacing your shift solenoid (aka valve bodie)

sounds mechanical. No reverse and no lower gears. If it were an electrical or computer control issue would default to second gear and still have reverse. Sorry, but I think you will have to pull the tranny on this one.
Good luck.

try fuel filter first, and see if there is any difference.

Most likely your shift cable (from the selector to the transmission) is worn, maybe needs lubrication. Mine had a similiar problem, but had to shift it into neutral and back up to get it in reverse. A new shift cable solved the problem.

It seems in the manual transmissions the first gear to go is 3rd gear..
signs of a failing trans