Also, he would want to be careful passing Gibralar, it was defended by 14 x 9.2 inch guns.
That also had the assistance of 11 x 6 inch and 7 x 4" guns. They couldn't sink the Goeben, but they could do a lot of damage to Breslau.
Were the straights mined to any significant extent? I believe they were later in the war, but it posed some technical challenges because of currents and it’s a major international waterway.
As far as I know there weren't mined to any extent whatsoever.
The Corps of Royal Engineers formed a submarine mining branch in 1863, but it was transferred to the Royal Navy in 1905 which (according to the History of the Corps of Royal Engineers which source I'm using) abolished a well organised system. AFAIK the transferred RE personnel became Royal Marines, but they were soon disbanded because Fisher thought that submarines were better for local defence.
While I'm at it this is the Garrison of Gibraltar in August 1914.
According to the Army Estimate 1914-15 the personnel establishment was 3,870 all ranks.
However, that's the maximum permitted strength. The actual strength would have been less than that.
According to the Army List, August 1914 the Troops in the Command were:
Arms
7 Royal Garrison Artillery companies:
Nos. 6, 9, 54 and 55 Companies (Northern Section).
Nos. 4, 7 & 8 Companies (Southern Section).
4 Fortress Companies, Royal Engineers (Nos. 1, 15, 32 and 45).
2 Infantry Battalions:
2nd Battalion, The Royal Scots Fusiliers.
2nd Battalion, The Due of Edinburgh's (Wiltshire Regiment).
Services
No. 39 Company, Army Service Corps.
No. 28 Company, Royal Army Medical Corps.
No. 9 Company (Gibraltar Section), Army Ordnance Corps