When -OH is converted to -OTs, the bond that forms is O-S (O-SO2C6H5CH3) and the bond that breaks is O-H.

No bonds to carbon are formed or broken.

Therefore the stereochemistry of the C-O bond is unchanged.

In practice, you will obtain a mixture that will require separation, and thus attempting to do so is impractical.
It is possible to selectively protect a primary alcohol through use of the extremely bulky triphenylmethyl (“trityl”) group.

HCl will form. In some cases, particularly with benzylic or tertiary alcohols, the O will then be protonated, resulting in an even better leaving group. Through either ionization (SN1 pathway) or direct displacement, substitution of OTs for Cl can then occur. The result is that alkyl chlorides are obtained, sometimes with scrambling of stereochemistry due to the SN1 pathway. To prevent this, 1 equivalent of base is used.
Related to this: https://pubs.acs.org/doi/abs/10.1021/jo00829a084

Because strong acid creates R-OH2+ , which is itself a strong acid. What if you wanted to use as your nucleophile the acetylide ion RCC (-) ? That’s a useful way of making a new carbon-carbon bond, right? Well, the acetylide ion is very basic. If acid is present, then the acetylide ion will just be protonated to give the alkyne, ,RCCH. No reaction.
So, in short, one good reason is that a leaving group like OTs or OMs is more compatible with strongly basic nucleophiles. I discuss this more in this post: https://www.masterorganicchemistry.com/2015/03/10/tosylates-and-mesylates/

Tertiary alcohols may be converted to tosylates and mesylates, yes.