I took a look at this puzzle, and some of the clues are very definitely not to my taste. However, the two clues that you quote are relatively sound.
Don't allow Doctor Love
Crossword custom and practice allows words to be capitalized in clues even when they are not proper nouns and are not at the start of a sentence, although setters try to avoid this where possible; the reverse (eg writing 'nice' when the wordplay refers to the French city) is considered unacceptable. The thinking behind this is that in some circumstances a word like 'love' can appear with a capital L (eg when it is at the start of a sentence), but the proper noun Nice can never be seen with a lower-case first letter. So 'Doctor Love' here is ok; note though that Chambers gives the 'vet' meaning of 'doctor' as exclusively North American, and the surface reading seems to make no sense - Don't allow Doctor Love to do what?
Like sea bed where horses run
It's a little tenuous, but Chambers gives the adjective 'sea' as meaning 'marine', and 'marine' as 'used at sea'. 'Cot' ('a swinging bed of canvas for naval officers') is indeed a bed used at sea.