Are there running gags per series in Top Gear?

• The Longest running gag is the Stig's introduction, which is always in the format of "Some say he [list of ridiculous claims] ... all we know is he's called The Stig.":
  • 'Some say he never blinks, and that he roams around the woods at night foraging for wolves. All we know is he’s called the Stig'
  • Some say that he thought Star Wars was a documentary, and that he recently pulled out of “I’m a Celebrity” because he’s frightened of trees, and Australia, and Koo Stark, and Ant, and Dec. All we know is he’s called the Stig'
• As mentioned in this question, during road trips the gang give each other inconvenient (to transport) gifts.
• Whenever travelling through hot countries, if any of the crew have air conditioning in their cars, the others will always break it (e.g. May in India and the US)
• They often reads a letter of complaint apparently from a fan called Mr. Needham which usually starts with "Dear so-called Top Gear" (or "Dear Top so-called Gear") and chastises them for doing ridiculous tests and challenges which have no bearing in the real world.
  • S08E03: Mr. Needham complains they never show sensible reasonably priced cars, Clarkson responds by showing a picture of a Vauxhall Corsa and immediately moves on.
  • S12E06: Mr. Needham asks if they've forgotten how to do 'proper tests, like in the old days', so Clarkson reviews the new Ford Fiesta and in a very deadpan manner begins listing the features of the car; 'There's room in the boot for a Zebra's head', 'There's enough room for the traditional 2.2 children', before eventually getting into a high speed chase through a shopping mall in scenes reminiscent of The Blues Brothers
  • S14E04: Clarkson reviews a Renaultsport Twingo 133 and begins testing silly features like 'Can you drive it upside down in a sewer?', 'Can Ross Kemp comfortably fit in the boot?' and 'What if you need to drive it on an Ice Rink?'

Yes, some of the series have series-contained running jokes. Wikipedia handily documents some of them on the Top Gear series-specific pages. Series 11 has:

The series included running jokes appearing in each episode: Clarkson showing an image he claims to have found on "the Internet" which is censored for the television broadcast; and, in the news section, May appearing to feign enthusiasm whilst presenting a brief news item about the Dacia Sandero with no further discussion before they move on to the next item.

Series 12 has:

The running joke regarding the Dacia Sandero continues from series 11, with the alteration that Clarkson is now the one making the announcement with feigned enthusiasm, to utter indifference from May (instead of vice versa). Each news segment also features one of the hosts wearing something unusual, like slippers or a garishly patterned shirt. The only reference to this is that one of the other hosts will, at some point in the segment, ask if they're wearing it for a bet.

Some of the episode summaries comment on running jokes as well.

Every so often they'll do part of a shoot on somebody else's track of which the location is 'supposedly' a secret to the general public. Of-course, first thing they do is explaining in excruciating detail where it's located.

For example, in Series 9 Episode 2 (where they turn a Reliant Robin into a Space Shuttle).

James: The launch site was a military base, whose whereabouts are a top secret.

Text on screen:
Clanburne Military Training Ground
53 miles N.W. of Newcastle
Just off the A68

Another example, in Series 10 Episode 7:

Clarkson: ... we were sent to the MIRA test track. Top secret place, it's where the motor industry tests all its secret new models. It's just outside Nuneaton, on the A5, between Hinckley and Everton, just before the A444. If you get to Fenny Drayton you've gone to far. You need to back up.

Of-course, the location of the (former) Motor Industry Research Association) is not a secret at all.

This gag had at this point been running for a couple of seasons already.

Another gag is James May getting lost half the time, illustrated in that same episode.

Richard Hammond was hypnotized to love 911s by Paul McKenna in 2004. Since that he has showed utter love for 911s.