DISCLAIMER: The following answers reflect my own knowledge and might not be 100% accurate. I am no dev or expert, just ur usual user.
1a. The prime benefit of rooting is having access to all the system partitions and files from within the rom (for both yourself and the applications you run). Thus you are enabled to make changes to the files that are normally locked for your personal use. This enables some applications to function which would normally not. (take titanium backup for example which is allowed (by rooting) to copy your system files and then restore them to a previous state if you encounter a file corruption or whatever). You can then via various apps overclock, uv, gps-fix and other system tweaks.
1b. by rooting and installing custom rom you indeed lose the ability to receive OTA updates from htc BUT you can find those roms on xda more probably before they would be available to you due to carrier specific delays and can install them manually. More than that most OTA updates are preceded by releases from the developers here on xda that are either leaked from htc, ported from other devices or built from scratch by our developers. For example, the desire S never received a sense 3.5 gingerbread update officialy as far as i know but there are GB Sense 3.5 ROMS available in the development section that were ported from other devices and which work flawlessly, IMHO better than stock ones being more stable, customizable, and packing huge functionality and aesthetic upgrades from the official ROM.
2. S-OFF: this unlocks your bootloader allowing you to flash a custom recovery like 4-EXT or CWM which allows you to do sub-rom operations (writing to the phones system partitions thus being alble to flash custom roms or restore a previous backup of your rom or do adb or in-recovery operations, all of these without loading the rom - it's like working in ms-dos for PC's). The advantages of a custom recovery are enormous. For example your situation: you can check whether your onboard microsd reader is hardware damaged or your rom encountered a corruption and can no longer handle the sdcard. Moreover you can rebuild the filesystem of your sdcard if it became corrupted in a way that is 100% android-compatible!
3. Yes you can. It is detailed in numerous threads in the general section of the forum.
4. Answered above and in previous posts in your thread: first it will enable you to test if it's a hardware or software failure, if it's software it will fix it via recovery.
If you understand these and wish to go on rooting and s-off read here:
You can find guides for s-offing both higher and lower than 2.0 hboot. If you are on >2.0 HBOOT i recommend the downgrade method.
: There are risks involved so be sure to read twice and follow the instructions to the word! I also recommend doing this process at 100% charged phone battery and on a laptop or on a UPS powered PC to avoid power blackouts. If you follow these rules you are 99.99% safe from bricking your phone.
PS: you do not need your sdcard to perform the s-off or root
PPS: it might be a good idea to post your hboot information here so i can provide further help.
Have fun and be safe!
Device: HTC ONE [M7]
History: Bosch COM509, Ericsson T20e, Sony Ericsson T310, Sony Ericsson T610, Nokia 6600, Nokia 6630, Nokia 7710, Nokia N72, Nokia N95, Nokia N97mini, HTC Desire S, HTC One S