Ask Mr. Smarty Plants

Yes, we don’t recommend Swedish Ivy or Asian Jasmine either. This is because it is our mission to increase the sustainable use and conservation of native wildflowers, plants and landscapes and both of those plants are non-native.  There are more than 350 species of Plectranthus spp. (Swedish ivy) that come from from Africa, Australia and Asia.  A few of them are named on this Univ. of Vermont page.  eHow states that “Swedish ivy is not native to Sweden and it is not technically ivy. It originated in Australia, but became a popular house plant in Sweden.”  I suspect it will not survive the winters in your area.  Similar statements apply to Trachelospermum asiaticum (Asian jasmine).  Here is a factsheet from NCSU about Trachelospermum asiaticum.  Asian Jasmine is native to Japan and Korea and can be quite invasive.

Mr Smarty Plants receives a lot of questions about whether a plant is toxic or not.  We tend to give a number of references and if a plant definitively shows in these references, then you can be sure it’s toxic; otherwise you can’t be sure whether or not it's poisonous.  Here are a few of these questions and answers; we didn’t find Swedish Ivy or Asian Jasmine listed in any of the sites.  One of the first questions to Mr Smarty Plants was a general request about plants poisonous to Dogs. Another question asked about pet-safe shrubs in McKinney and whether Coral Honeysuckle is poisonous to dogs.  These have good lists of references, I didn't find your plants listed in any of them, but you should check yourself to see if they appear on any of the lists.  Within all these references, one of the best "poisonous to dogs" sites is from the ASPCA and lists both poisonous plants and safe plants.  The listing is by common name but you can always use the "Find" search to look up scientific name--or the common name for that matter.  

So, what would I recommend?  There is a great search facility on this website.  It’s called “Recommended Species” and you can look at North-Central Texas and select “0-1 feet” and you’ll get several recommendations to consider.  Of this list, I liked Glandularia bipinnatifida var. bipinnatifida (Prairie verbena), Dichondra argentea (Silver ponyfoot), Viola missouriensis (Missouri violet), Chasmanthium latifolium (Inland sea oats), Phlox [Phlox pilosa (Downy phlox) or Phlox divaricata (Wild blue phlox)] and Oenothera speciosa (Pink evening primrose).  All are tolerant of at least partial shade and will grow well in your area.  There were 40 recommended species of herbs and 10 that were 0-1 feet tall.

We also recommended two other species when asked a similar question where we explored groundcover for Possum Kingdom.  These are Phyla nodiflora (Texas frogfruit) or Calyptocarpus vialis (Straggler daisy).  The Frogfruit is expected to grow to 3"-6" and the Straggler daisy to 6"-12" at the maximum.  We have these in our backyard and our dogs have munched the Straggler Daisy, so I expect they will be OK!