Hi Tracy,
Suggest that you take a look at the
Albena website - on the main page are links to the various sports & activities, including horse riding.
Golf courses - no, some are under development in the Northern Black Sea coastal area near Albena but, as far as I'm aware, none are yet fully functional. The nearest that you'll find is likely to be a mini/crazy golf layout in the resort itself - there's one just off the main promenade area, from memory.
Cycling - can't say that I recall seeing cycle hire facilities but I wasn't looking. There are almost certain to be some and cycling is mentioned in the sports links on the website. There are certainly multi-seat pedalled buggies flying around the resort most of the time.
There is a bus station at the entrance to the resort, with regular buses to the cities of
Varna &
Dobrich, as well as nearby towns, such as
Balchik. A bus ride to Dobrich would give you a view of some lovely forested countryside as you climb up to the Dobrudzha plateau, fields of sunflowers & other crops on the plateau itself, as well as passing through typical Bulgarian villages like Obrochiste, Tsarkva, Batovo, Slaveevo and Branishte en route to Dobrich, where eating, shopping, etc., are much cheaper than in the resorts - you can also get some idea of 'real' life in Bulgaria. The adult fare to Dobrich was 4 Leva (around £1.40) the last time that I made the trip (last September).
As celiajan says, I would personally not offer gifts to any children - until or unless you make particular friends of the parents first.
Baldur