Torn boot/seal on underside, difficulty shifting.

Recently I bought a Jimny 2006 1.3L with manual transmission, without completely checking the underside of the car. So far the car fortunately seems to run fine.

Anyone want to give advise about the following issues?:

1. On the underside of the car a seal or rubber boot is broken. It seems to be part of the transmission as far as my limited knowledge goes. Does anybody recognize the seal/boot on the picture and can tell my the name of the part? The local Suzuki dealer doesn't seem confident in naming the correct part. The manual of the Jimny makes notice of one seal on the underside called 'boot', situated next to the 'gear shift lever case' and above the 'extension rod'.

2. The shifting to first and second gear goes rather stiffly. Sometimes engaging into second gear requires much force. (although it isn't continuously present). The shifting seems to go easier without the engine running, but I'm not sure considering the low frequency of shifting with the engine off...

Maybe problem 1 and 2 relate to each other.
What other parts should I consider in the stiff shifting? (the dealer advised to renew the transmission cover assy, disc and release bearing)

Thanks in advance and greetings,
David