I'm sure the b@st@rds change the dimensions every few years.
1 - Sounds like "ogive" or "half round"
2 - Find a brand name or such on it
3 - If it is black consider binning & buying white if south facing
Plastic expands significantly from 10oC to 35oC. Over 10 metres that expansion is significant and must be absorbed by couplers whilst still remaining watertight. All couplers should be fixed to ensure they do not become low spots, use at least the number of brackets as specified by the manufacturer. More couplers (ironically) can reduce the expansion range each coupler on that run has to accommodate.
4 - I suspect "deep-U-trough" is better than "half-round"
With "half-round" you have to get the fall exactly right to avoid it overflowing in sudden rainfall, even then it can still shed water due to moss accumulation or wind interference. Get the fall wrong and you can have significant overflow which brickwork simply can not handle, UK houses do not follow German & Dutch where the eaves extend some way out protecting both brickwork & providing a longer lived roof.
With "deep-U-trough" the depth is much more forgiving w.r.t. fall. Downsides might be that it is narrower, so height relative to flashing may be more sensitive and flow rate may be lower providing greater opportunity for moss accumulation (although the body of water behind it will be higher so potentially persuading it on its way).
Just depends on your rainfall pattern.
5 - If your gutter flashing is rotten, replace it
Very carefully lift up each tile with wooden wedges. Slide under black DPC underneath the existing felt so that the DPC hangs into the guttering. Most tiles are nailed every 5, but the bottom row can be all nailed (it varies) so don't get too enthusiastic. Depending on the direction of the prevailing wind this can greatly reduce gutter-miss, soffit-soaking & wall soaking (even back into the cavity).
If your neighbours guttering must fix to yours, welcome to misery, it will probably be a different size. Many facing bricks are lousy at handling guttering leaks, hence the sudden move to "cavity wall trays" with weep vents to help.