What does the word Mountain mean on these?

Mountain was a type of fortified wine from Malaga in Spain. It was a very common drink in eighteenth-century Britain. The wine was sweet and similar to sherry. Since it was a staple of the Georgian cellar, silversmiths made thousands of labels for it. You see them regularly among early antique silver decanter labels.

I have a few silver bottle tickets with no hallmarks at all, just a maker's stamp on the back, why were they not fully assayed by the office?

Small silver items were exempt from hallmarking for much of the Georgian period. The assay office rules excluded wares weighing under a certain amount. Decanter labels fell into this category. Many makers just struck their personal mark to show who made it. Some did not even bother with that. Assaying cost time and money, so silversmiths skipped the process legally when making lightweight goods.

Can I clean tarnished engraved lettering?

Use a soft silver cloth rather than liquid dips. Harsh chemicals strip the dark tarnish out of the engraved letters. That dark oxidisation actually helps the word stand out against the polished silver. If you scrub it all away, the spirit name becomes very difficult to read.

Some of the silver chains on my older bottle labels are very long and slip right over the shoulders of the decanter, is this the original length?

Chains often get replaced or altered over the centuries. A standard length usually measures around five or six inches. This allows the label to hang neatly on the upper neck. If the chain slides down the body of the bottle, someone likely swapped it. People often repurposed pocket watch chains or necklace fragments when the original silver wire links broke.