Both problems are probably related to the nut on the transmission output shaft being loose. Tightening the nut is not a trivial job, but many have done it. The speedo drive gear is clamped by the pressure of the nut, if the nut is loose the speedo drive gear can spin free on the shaft and you get erratic readings. The same nut also holds first and second gear in place, and a loose nut allows excessive end-play which (if you're lucky) is the only cause of the grinding. Tightening the nut may solve both problems.
If you're not so lucky, the excessive end-play may have allowed a flat key inside the transmission to disengage from a slot it fits in, in which case the trans must be disassembled to fix. It's also possible the second gear synchro ring is shot.
The nut is a slotted nut, requires a special socket, which many members have made themselves. The driveshaft must be disconnected and flex disk removed, along with the transmission support plate and rear motor mount to get to the nut. Things to replace while doing this job are:
rear transmission seal
flex disk and all nuts/bots (they come in the kit)
shifter bushings
rear motor mount if necessary.
This is sort of double the work, but you could disassemble everything, tighten the nut (I use loctite on it), and reassemble TEMPORARILY with the old flex disk and then test drive it a day or two to see if that fixed it. Then go back in there and replace the old flex disk with the new one. This reflects a strong bias on my part to never reuse the flex disk nuts and bolts, because if they come loose you have a good chance of chewing up the driveshaft and/or trans output flange, which gets very expensive.
Do some searches on words like 'transmission' 'output' 'slotted' 'nut' and similar, you'll find lots of info on this.
Good luck!
George Davis
'69 280 SL Euro manual